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Theory of Computing Report

Thursday, April 24

Symmetric Proofs in the Ideal Proof System

from arXiv: Computational Complexity

Authors: Anuj Dawar, Erich Grädel, Leon Kullmann, Benedikt Pago

We consider the Ideal Proof System (IPS) introduced by Grochow and Pitassi and pose the question of which tautologies admit symmetric proofs, and of what complexity. The symmetry requirement in proofs is inspired by recent work establishing lower bounds in other symmetric models of computation. We link the existence of symmetric IPS proofs to the expressive power of logics such as fixed-point logic with counting and Choiceless Polynomial Time, specifically regarding the graph isomorphism problem. We identify relationships and tradeoffs between the symmetry of proofs and other parameters of IPS proofs such as size, degree and linearity. We study these on a number of standard families of tautologies from proof complexity and finite model theory such as the pigeonhole principle, the subset sum problem and the Cai-F\"urer-Immerman graphs, exhibiting non-trivial upper bounds on the size of symmetric IPS proofs.

Authors: Anuj Dawar, Erich Grädel, Leon Kullmann, Benedikt Pago

We consider the Ideal Proof System (IPS) introduced by Grochow and Pitassi and pose the question of which tautologies admit symmetric proofs, and of what complexity. The symmetry requirement in proofs is inspired by recent work establishing lower bounds in other symmetric models of computation. We link the existence of symmetric IPS proofs to the expressive power of logics such as fixed-point logic with counting and Choiceless Polynomial Time, specifically regarding the graph isomorphism problem. We identify relationships and tradeoffs between the symmetry of proofs and other parameters of IPS proofs such as size, degree and linearity. We study these on a number of standard families of tautologies from proof complexity and finite model theory such as the pigeonhole principle, the subset sum problem and the Cai-F\"urer-Immerman graphs, exhibiting non-trivial upper bounds on the size of symmetric IPS proofs.

Online and feasible presentability: from trees to modal algebras

from arXiv: Computational Complexity

Authors: Nikolay Bazhenov, Dariusz Kalociński, Michał Wrocławski

We investigate whether every computable member of a given class of structures admits a fully primitive recursive (also known as punctual) or fully P-TIME copy. A class with this property is referred to as punctually robust or P-TIME robust, respectively. We present both positive and negative results for structures corresponding to well-known representations of trees, such as binary trees, ordered trees, sequential (or prefix) trees, and partially ordered (poset) trees. A corollary of one of our results on trees is that semilattices and lattices are not punctually robust. In the main result of the paper, we demonstrate that, unlike Boolean algebras, modal algebras - that is, Boolean algebras with modality - are not punctually robust. The question of whether distributive lattices are punctually robust remains open. The paper contributes to a decades-old program on effective and feasible algebra, which has recently gained momentum due to rapid developments in punctual structure theory and its connections to online presentations of structures.

Authors: Nikolay Bazhenov, Dariusz Kalociński, Michał Wrocławski

We investigate whether every computable member of a given class of structures admits a fully primitive recursive (also known as punctual) or fully P-TIME copy. A class with this property is referred to as punctually robust or P-TIME robust, respectively. We present both positive and negative results for structures corresponding to well-known representations of trees, such as binary trees, ordered trees, sequential (or prefix) trees, and partially ordered (poset) trees. A corollary of one of our results on trees is that semilattices and lattices are not punctually robust. In the main result of the paper, we demonstrate that, unlike Boolean algebras, modal algebras - that is, Boolean algebras with modality - are not punctually robust. The question of whether distributive lattices are punctually robust remains open. The paper contributes to a decades-old program on effective and feasible algebra, which has recently gained momentum due to rapid developments in punctual structure theory and its connections to online presentations of structures.

Key-agreement exists if and only if the "interactive vs non interactive Kolmogorov problem" is not in ioBPP: a short proof

from arXiv: Computational Complexity

Authors: Bruno Bauwens, Bruno Loff

Ball, Liu, Mazor and Pass proved that the existence of key-agreement protocols is equivalent to the hardness of a certain problem about interactive Kolmogorov complexity. We generalize the statement and give a short proof of the difficult implication.

Authors: Bruno Bauwens, Bruno Loff

Ball, Liu, Mazor and Pass proved that the existence of key-agreement protocols is equivalent to the hardness of a certain problem about interactive Kolmogorov complexity. We generalize the statement and give a short proof of the difficult implication.

Drainability and Fillability of Polyominoes in Diverse Models of Global Control

from arXiv: Computational Geometry

Authors: Sándor P. Fekete, Peter Kramer, Jan-Marc Reinhardt, Christian Rieck, Christian Scheffer

Tilt models offer intuitive and clean definitions of complex systems in which particles are influenced by global control commands. Despite a wide range of applications, there has been almost no theoretical investigation into the associated issues of filling and draining geometric environments. This is partly because a globally controlled system (i.e., passive matter) exhibits highly complex behavior that cannot be locally restricted. Thus, there is a strong need for theoretical studies that investigate these models both (1) in terms of relative power to each other, and (2) from a complexity theory perspective. In this work, we provide (1) general tools for comparing and contrasting different models of global control, and (2) both complexity and algorithmic results on filling and draining.

Authors: Sándor P. Fekete, Peter Kramer, Jan-Marc Reinhardt, Christian Rieck, Christian Scheffer

Tilt models offer intuitive and clean definitions of complex systems in which particles are influenced by global control commands. Despite a wide range of applications, there has been almost no theoretical investigation into the associated issues of filling and draining geometric environments. This is partly because a globally controlled system (i.e., passive matter) exhibits highly complex behavior that cannot be locally restricted. Thus, there is a strong need for theoretical studies that investigate these models both (1) in terms of relative power to each other, and (2) from a complexity theory perspective. In this work, we provide (1) general tools for comparing and contrasting different models of global control, and (2) both complexity and algorithmic results on filling and draining.

Hitting and Covering Affine Families of Convex Polyhedra, with Applications to Robust Optimization

from arXiv: Computational Geometry

Authors: Jean Cardinal, Xavier Goaoc, Sarah Wajsbrot

Geometric hitting set problems, in which we seek a smallest set of points that collectively hit a given set of ranges, are ubiquitous in computational geometry. Most often, the set is discrete and is given explicitly. We propose new variants of these problems, dealing with continuous families of convex polyhedra, and show that they capture decision versions of the two-level finite adaptability problem in robust optimization. We show that these problems can be solved in strongly polynomial time when the size of the hitting/covering set and the dimension of the polyhedra and the parameter space are constant. We also show that the hitting set problem can be solved in strongly quadratic time for one-parameter families of convex polyhedra in constant dimension. This leads to new tractability results for finite adaptability that are the first ones with so-called left-hand-side uncertainty, where the underlying problem is non-linear.

Authors: Jean Cardinal, Xavier Goaoc, Sarah Wajsbrot

Geometric hitting set problems, in which we seek a smallest set of points that collectively hit a given set of ranges, are ubiquitous in computational geometry. Most often, the set is discrete and is given explicitly. We propose new variants of these problems, dealing with continuous families of convex polyhedra, and show that they capture decision versions of the two-level finite adaptability problem in robust optimization. We show that these problems can be solved in strongly polynomial time when the size of the hitting/covering set and the dimension of the polyhedra and the parameter space are constant. We also show that the hitting set problem can be solved in strongly quadratic time for one-parameter families of convex polyhedra in constant dimension. This leads to new tractability results for finite adaptability that are the first ones with so-called left-hand-side uncertainty, where the underlying problem is non-linear.

Hardness of Median and Center in the Ulam Metric

from arXiv: Data Structures and Algorithms

Authors: Nick Fischer, Elazar Goldenberg, Mursalin Habib, C. S. Karthik

The classical rank aggregation problem seeks to combine a set X of n permutations into a single representative "consensus" permutation. In this paper, we investigate two fundamental rank aggregation tasks under the well-studied Ulam metric: computing a median permutation (which minimizes the sum of Ulam distances to X) and computing a center permutation (which minimizes the maximum Ulam distance to X) in two settings. $\bullet$ Continuous Setting: In the continuous setting, the median/center is allowed to be any permutation. It is known that computing a center in the Ulam metric is NP-hard and we add to this by showing that computing a median is NP-hard as well via a simple reduction from the Max-Cut problem. While this result may not be unexpected, it had remained elusive until now and confirms a speculation by Chakraborty, Das, and Krauthgamer [SODA '21]. $\bullet$ Discrete Setting: In the discrete setting, the median/center must be a permutation from the input set. We fully resolve the fine-grained complexity of the discrete median and discrete center problems under the Ulam metric, proving that the naive $\widetilde{O}(n^2 L)$-time algorithm (where L is the length of the permutation) is conditionally optimal. This resolves an open problem raised by Abboud, Bateni, Cohen-Addad, Karthik C. S., and Seddighin [APPROX '23]. Our reductions are inspired by the known fine-grained lower bounds for similarity measures, but we face and overcome several new highly technical challenges.

Authors: Nick Fischer, Elazar Goldenberg, Mursalin Habib, C. S. Karthik

The classical rank aggregation problem seeks to combine a set X of n permutations into a single representative "consensus" permutation. In this paper, we investigate two fundamental rank aggregation tasks under the well-studied Ulam metric: computing a median permutation (which minimizes the sum of Ulam distances to X) and computing a center permutation (which minimizes the maximum Ulam distance to X) in two settings. $\bullet$ Continuous Setting: In the continuous setting, the median/center is allowed to be any permutation. It is known that computing a center in the Ulam metric is NP-hard and we add to this by showing that computing a median is NP-hard as well via a simple reduction from the Max-Cut problem. While this result may not be unexpected, it had remained elusive until now and confirms a speculation by Chakraborty, Das, and Krauthgamer [SODA '21]. $\bullet$ Discrete Setting: In the discrete setting, the median/center must be a permutation from the input set. We fully resolve the fine-grained complexity of the discrete median and discrete center problems under the Ulam metric, proving that the naive $\widetilde{O}(n^2 L)$-time algorithm (where L is the length of the permutation) is conditionally optimal. This resolves an open problem raised by Abboud, Bateni, Cohen-Addad, Karthik C. S., and Seddighin [APPROX '23]. Our reductions are inspired by the known fine-grained lower bounds for similarity measures, but we face and overcome several new highly technical challenges.

Approximating Optimal Labelings for Temporal Connectivity

from arXiv: Data Structures and Algorithms

Authors: Daniele Carnevale, Gianlorenzo D'Angelo, Martin Olsen

In a temporal graph the edge set dynamically changes over time according to a set of time-labels associated with each edge that indicates at which time-steps the edge is available. Two vertices are connected if there is a path connecting them in which the edges are traversed in increasing order of their labels. We study the problem of scheduling the availability time of the edges of a temporal graph in such a way that all pairs of vertices are connected within a given maximum allowed time $a$ and the overall number of labels is minimized. The problem, known as \emph{Minimum Aged Labeling} (MAL), has several applications in logistics, distribution scheduling, and information spreading in social networks, where carefully choosing the time-labels can significantly reduce infrastructure costs, fuel consumption, or greenhouse gases. The problem MAL has previously been proved to be NP-complete on undirected graphs and \APX-hard on directed graphs. In this paper, we extend our knowledge on the complexity and approximability of MAL in several directions. We first show that the problem cannot be approximated within a factor better than $O(\log n)$ when $a\geq 2$, unless $\text{P} = \text{NP}$, and a factor better than $2^{\log ^{1-\epsilon} n}$ when $a\geq 3$, unless $\text{NP}\subseteq \text{DTIME}(2^{\text{polylog}(n)})$, where $n$ is the number of vertices in the graph. Then we give a set of approximation algorithms that, under some conditions, almost match these lower bounds. In particular, we show that the approximation depends on a relation between $a$ and the diameter of the input graph. We further establish a connection with a foundational optimization problem on static graphs called \emph{Diameter Constrained Spanning Subgraph} (DCSS) and show that our hardness results also apply to DCSS.

Authors: Daniele Carnevale, Gianlorenzo D'Angelo, Martin Olsen

In a temporal graph the edge set dynamically changes over time according to a set of time-labels associated with each edge that indicates at which time-steps the edge is available. Two vertices are connected if there is a path connecting them in which the edges are traversed in increasing order of their labels. We study the problem of scheduling the availability time of the edges of a temporal graph in such a way that all pairs of vertices are connected within a given maximum allowed time $a$ and the overall number of labels is minimized. The problem, known as \emph{Minimum Aged Labeling} (MAL), has several applications in logistics, distribution scheduling, and information spreading in social networks, where carefully choosing the time-labels can significantly reduce infrastructure costs, fuel consumption, or greenhouse gases. The problem MAL has previously been proved to be NP-complete on undirected graphs and \APX-hard on directed graphs. In this paper, we extend our knowledge on the complexity and approximability of MAL in several directions. We first show that the problem cannot be approximated within a factor better than $O(\log n)$ when $a\geq 2$, unless $\text{P} = \text{NP}$, and a factor better than $2^{\log ^{1-\epsilon} n}$ when $a\geq 3$, unless $\text{NP}\subseteq \text{DTIME}(2^{\text{polylog}(n)})$, where $n$ is the number of vertices in the graph. Then we give a set of approximation algorithms that, under some conditions, almost match these lower bounds. In particular, we show that the approximation depends on a relation between $a$ and the diameter of the input graph. We further establish a connection with a foundational optimization problem on static graphs called \emph{Diameter Constrained Spanning Subgraph} (DCSS) and show that our hardness results also apply to DCSS.

Graph modification of bounded size to minor-closed classes as fast as vertex deletion

from arXiv: Data Structures and Algorithms

Authors: Laure Morelle, Ignasi Sau, Dimitrios M. Thilikos

A replacement action is a function $\mathcal{L}$ that maps each graph $H$ to a collection of graphs of size at most $|V(H)|$. Given a graph class $\mathcal{H}$, we consider a general family of graph modification problems, called $\mathcal{L}$-Replacement to $\mathcal{H}$, where the input is a graph $G$ and the question is whether it is possible to replace some induced subgraph $H_1$ of $G$ on at most $k$ vertices by a graph $H_2$ in $\mathcal{L}(H_1)$ so that the resulting graph belongs to $\mathcal{H}$. $\mathcal{L}$-Replacement to $\mathcal{H}$ can simulate many graph modification problems including vertex deletion, edge deletion/addition/edition/contraction, vertex identification, subgraph complementation, independent set deletion, (induced) matching deletion/contraction, etc. We present two algorithms. The first one solves $\mathcal{L}$-Replacement to $\mathcal{H}$ in time $2^{{\rm poly}(k)}\cdot |V(G)|^2$ for every minor-closed graph class $\mathcal{H}$, where {\rm poly} is a polynomial whose degree depends on $\mathcal{H}$, under a mild technical condition on $\mathcal{L}$. This generalizes the results of Morelle, Sau, Stamoulis, and Thilikos [ICALP 2020, ICALP 2023] for the particular case of Vertex Deletion to $\mathcal{H}$ within the same running time. Our second algorithm is an improvement of the first one when $\mathcal{H}$ is the class of graphs embeddable in a surface of Euler genus at most $g$ and runs in time $2^{\mathcal{O}(k^{9})}\cdot |V(G)|^2$, where the $\mathcal{O}(\cdot)$ notation depends on $g$. To the best of our knowledge, these are the first parameterized algorithms with a reasonable parametric dependence for such a general family of graph modification problems to minor-closed classes.

Authors: Laure Morelle, Ignasi Sau, Dimitrios M. Thilikos

A replacement action is a function $\mathcal{L}$ that maps each graph $H$ to a collection of graphs of size at most $|V(H)|$. Given a graph class $\mathcal{H}$, we consider a general family of graph modification problems, called $\mathcal{L}$-Replacement to $\mathcal{H}$, where the input is a graph $G$ and the question is whether it is possible to replace some induced subgraph $H_1$ of $G$ on at most $k$ vertices by a graph $H_2$ in $\mathcal{L}(H_1)$ so that the resulting graph belongs to $\mathcal{H}$. $\mathcal{L}$-Replacement to $\mathcal{H}$ can simulate many graph modification problems including vertex deletion, edge deletion/addition/edition/contraction, vertex identification, subgraph complementation, independent set deletion, (induced) matching deletion/contraction, etc. We present two algorithms. The first one solves $\mathcal{L}$-Replacement to $\mathcal{H}$ in time $2^{{\rm poly}(k)}\cdot |V(G)|^2$ for every minor-closed graph class $\mathcal{H}$, where {\rm poly} is a polynomial whose degree depends on $\mathcal{H}$, under a mild technical condition on $\mathcal{L}$. This generalizes the results of Morelle, Sau, Stamoulis, and Thilikos [ICALP 2020, ICALP 2023] for the particular case of Vertex Deletion to $\mathcal{H}$ within the same running time. Our second algorithm is an improvement of the first one when $\mathcal{H}$ is the class of graphs embeddable in a surface of Euler genus at most $g$ and runs in time $2^{\mathcal{O}(k^{9})}\cdot |V(G)|^2$, where the $\mathcal{O}(\cdot)$ notation depends on $g$. To the best of our knowledge, these are the first parameterized algorithms with a reasonable parametric dependence for such a general family of graph modification problems to minor-closed classes.

Traffic-Oblivious Multi-Commodity Flow Network Design

from arXiv: Data Structures and Algorithms

Authors: Markus Chimani, Max Ilsen

We consider the Minimum Multi-Commodity Flow Subgraph (MMCFS) problem: given a directed graph $G$ with edge capacities $\mathit{cap}$ and a retention ratio $\alpha\in(0,1)$, find an edge-wise minimum subgraph $G' \subseteq G$ such that for all traffic matrices $T$ routable in $G$ using a multi-commodity flow, $\alpha\cdot T$ is routable in $G'$. This natural yet novel problem is motivated by recent research that investigates how the power consumption in backbone computer networks can be reduced by turning off connections during times of low demand without compromising the quality of service. Since the actual traffic demands are generally not known beforehand, our approach must be traffic-oblivious, i.e., work for all possible sets of simultaneously routable traffic demands in the original network. In this paper we present the problem, relate it to other known problems in literature, and show several structural results, including a reformulation, maximum possible deviations from the optimum, and NP-hardness (as well as a certain inapproximability) already on very restricted instances. The most significant contribution is a tight $\max(\frac{1}{\alpha}, 2)$-approximation based on an algorithmically surprisingly simple LP-rounding scheme.

Authors: Markus Chimani, Max Ilsen

We consider the Minimum Multi-Commodity Flow Subgraph (MMCFS) problem: given a directed graph $G$ with edge capacities $\mathit{cap}$ and a retention ratio $\alpha\in(0,1)$, find an edge-wise minimum subgraph $G' \subseteq G$ such that for all traffic matrices $T$ routable in $G$ using a multi-commodity flow, $\alpha\cdot T$ is routable in $G'$. This natural yet novel problem is motivated by recent research that investigates how the power consumption in backbone computer networks can be reduced by turning off connections during times of low demand without compromising the quality of service. Since the actual traffic demands are generally not known beforehand, our approach must be traffic-oblivious, i.e., work for all possible sets of simultaneously routable traffic demands in the original network. In this paper we present the problem, relate it to other known problems in literature, and show several structural results, including a reformulation, maximum possible deviations from the optimum, and NP-hardness (as well as a certain inapproximability) already on very restricted instances. The most significant contribution is a tight $\max(\frac{1}{\alpha}, 2)$-approximation based on an algorithmically surprisingly simple LP-rounding scheme.

From Theory to Practice: Engineering Approximation Algorithms for Dynamic Orientation

from arXiv: Data Structures and Algorithms

Authors: Ernestine Großmann, Ivor van der Hoog, Henrik Reinstädtler, Eva Rotenberg, Christian Schulz, Juliette Vlieghe

Dynamic graph algorithms have seen significant theoretical advancements, but practical evaluations often lag behind. This work bridges the gap between theory and practice by engineering and empirically evaluating recently developed approximation algorithms for dynamically maintaining graph orientations. We comprehensively describe the underlying data structures, including efficient bucketing techniques and round-robin updates. Our implementation has a natural parameter $\lambda$, which allows for a trade-off between algorithmic efficiency and the quality of the solution. In the extensive experimental evaluation, we demonstrate that our implementation offers a considerable speedup. Using different quality metrics, we show that our implementations are very competitive and can outperform previous methods. Overall, our approach solves more instances than other methods while being up to 112 times faster on instances that are solvable by all methods compared.

Authors: Ernestine Großmann, Ivor van der Hoog, Henrik Reinstädtler, Eva Rotenberg, Christian Schulz, Juliette Vlieghe

Dynamic graph algorithms have seen significant theoretical advancements, but practical evaluations often lag behind. This work bridges the gap between theory and practice by engineering and empirically evaluating recently developed approximation algorithms for dynamically maintaining graph orientations. We comprehensively describe the underlying data structures, including efficient bucketing techniques and round-robin updates. Our implementation has a natural parameter $\lambda$, which allows for a trade-off between algorithmic efficiency and the quality of the solution. In the extensive experimental evaluation, we demonstrate that our implementation offers a considerable speedup. Using different quality metrics, we show that our implementations are very competitive and can outperform previous methods. Overall, our approach solves more instances than other methods while being up to 112 times faster on instances that are solvable by all methods compared.

Sorting as Gradient Flow on the Permutohedron

from arXiv: Data Structures and Algorithms

Authors: Jonathan Landers

We investigate how sorting algorithms efficiently overcome the exponential size of the permutation space. Our main contribution is a new continuous-time formulation of sorting as a gradient flow on the permutohedron, yielding an independent proof of the classical $\Omega(n \log n)$ lower bound for comparison-based sorting. This formulation reveals how exponential contraction of disorder occurs under simple geometric dynamics. In support of this analysis, we present algebraic, combinatorial, and geometric perspectives, including decision-tree arguments and linear constraints on the permutohedron. The idea that efficient sorting arises from structure-guided logarithmic reduction offers a unifying lens for how comparisons tame exponential spaces. These observations connect to broader questions in theoretical computer science, such as whether the existence of structure can explain why certain computational problems permit efficient solutions.

Authors: Jonathan Landers

We investigate how sorting algorithms efficiently overcome the exponential size of the permutation space. Our main contribution is a new continuous-time formulation of sorting as a gradient flow on the permutohedron, yielding an independent proof of the classical $\Omega(n \log n)$ lower bound for comparison-based sorting. This formulation reveals how exponential contraction of disorder occurs under simple geometric dynamics. In support of this analysis, we present algebraic, combinatorial, and geometric perspectives, including decision-tree arguments and linear constraints on the permutohedron. The idea that efficient sorting arises from structure-guided logarithmic reduction offers a unifying lens for how comparisons tame exponential spaces. These observations connect to broader questions in theoretical computer science, such as whether the existence of structure can explain why certain computational problems permit efficient solutions.

Streaming algorithms for products of probabilities

from arXiv: Data Structures and Algorithms

Authors: Markus Lohrey, Leon Rische, Louisa Seelbach Benkner, Julio Xochitemol

We consider streaming algorithms for approximating a product of input probabilities up to multiplicative error of $1-\epsilon$. It is shown that every randomized streaming algorithm for this problem needs space $\Omega(\log n + \log b - \log \epsilon) - \mathcal{O}(1)$, where $n$ is length of the input stream and $b$ is the bit length of the input numbers. This matches an upper bound from Alur et al.~up to a constant multiplicative factor. Moreover, we consider the threshold problem, where it is asked whether the product of the input probabilities is below a given threshold. It is shown that every randomized streaming algorithm for this problem needs space $\Omega(n \cdot b)$.

Authors: Markus Lohrey, Leon Rische, Louisa Seelbach Benkner, Julio Xochitemol

We consider streaming algorithms for approximating a product of input probabilities up to multiplicative error of $1-\epsilon$. It is shown that every randomized streaming algorithm for this problem needs space $\Omega(\log n + \log b - \log \epsilon) - \mathcal{O}(1)$, where $n$ is length of the input stream and $b$ is the bit length of the input numbers. This matches an upper bound from Alur et al.~up to a constant multiplicative factor. Moreover, we consider the threshold problem, where it is asked whether the product of the input probabilities is below a given threshold. It is shown that every randomized streaming algorithm for this problem needs space $\Omega(n \cdot b)$.

Estimating Random-Walk Probabilities in Directed Graphs

from arXiv: Data Structures and Algorithms

Authors: Christian Bertram, Mads Vestergaard Jensen, Mikkel Thorup, Hanzhi Wang, Shuyi Yan

We study discounted random walks in a directed graph. In each vertex, the walk will either terminate with some probability $\alpha$, or continue to a random out-neighbor. We are interested in the probability $\pi(s,t)$ that such a random walk starting in $s$ ends in $t$. We wish to, with constant probability, estimate $\pi(s, t)$ within a constant relative error, unless $\pi(s, t) < \delta$ for some given threshold $\delta$. The current status is as follows. Algorithms with worst-case running time $\tilde O(m)$ and $O(1/\delta)$ are known. A more complicated algorithm is known, which does not perform better in the worst case, but for the average running time over all $n$ possible targets $t$, it achieves an alternative bound of $O(\sqrt{d/\delta})$. All the above algorithms assume query access to the adjacency list of a node. On the lower bound side, the best-known lower bound for the worst case is $\Omega(n^{1/2}m^{1/4})$ with $\delta \leq 1/(n^{1/2}m^{1/4})$, and for the average case it is $\Omega(\sqrt{n})$ with $\delta \leq 1/n$. This leaves substantial polynomial gaps in both cases. In this paper, we show that the above upper bounds are tight across all parameters $n$, $m$ and $\delta$. We show that the right bound is $\tilde\Theta(\min\{m, 1/\delta\})$ for the worst case, and $\tilde\Theta(\min\{m, \sqrt{d/\delta}, 1/\delta\})$ for the average case. We also consider some additional graph queries from the literature. One allows checking whether there is an edge from $u$ to $v$ in constant time. Another allows access to the adjacency list of $u$ sorted by out-degree. We prove that none of these access queries help in the worst case, but if we have both of them, we get an average-case bound of $\tilde \Theta(\min\{m,\sqrt{d/\delta}, (1/\delta)^{2/3}\})$.

Authors: Christian Bertram, Mads Vestergaard Jensen, Mikkel Thorup, Hanzhi Wang, Shuyi Yan

We study discounted random walks in a directed graph. In each vertex, the walk will either terminate with some probability $\alpha$, or continue to a random out-neighbor. We are interested in the probability $\pi(s,t)$ that such a random walk starting in $s$ ends in $t$. We wish to, with constant probability, estimate $\pi(s, t)$ within a constant relative error, unless $\pi(s, t) < \delta$ for some given threshold $\delta$. The current status is as follows. Algorithms with worst-case running time $\tilde O(m)$ and $O(1/\delta)$ are known. A more complicated algorithm is known, which does not perform better in the worst case, but for the average running time over all $n$ possible targets $t$, it achieves an alternative bound of $O(\sqrt{d/\delta})$. All the above algorithms assume query access to the adjacency list of a node. On the lower bound side, the best-known lower bound for the worst case is $\Omega(n^{1/2}m^{1/4})$ with $\delta \leq 1/(n^{1/2}m^{1/4})$, and for the average case it is $\Omega(\sqrt{n})$ with $\delta \leq 1/n$. This leaves substantial polynomial gaps in both cases. In this paper, we show that the above upper bounds are tight across all parameters $n$, $m$ and $\delta$. We show that the right bound is $\tilde\Theta(\min\{m, 1/\delta\})$ for the worst case, and $\tilde\Theta(\min\{m, \sqrt{d/\delta}, 1/\delta\})$ for the average case. We also consider some additional graph queries from the literature. One allows checking whether there is an edge from $u$ to $v$ in constant time. Another allows access to the adjacency list of $u$ sorted by out-degree. We prove that none of these access queries help in the worst case, but if we have both of them, we get an average-case bound of $\tilde \Theta(\min\{m,\sqrt{d/\delta}, (1/\delta)^{2/3}\})$.

Improved Streaming Edge Coloring

from arXiv: Data Structures and Algorithms

Authors: Shiri Chechik, Hongyi Chen, Tianyi Zhang

Given a graph, an edge coloring assigns colors to edges so that no pairs of adjacent edges share the same color. We are interested in edge coloring algorithms under the W-streaming model. In this model, the algorithm does not have enough memory to hold the entire graph, so the edges of the input graph are read from a data stream one by one in an unknown order, and the algorithm needs to print a valid edge coloring in an output stream. The performance of the algorithm is measured by the amount of space and the number of different colors it uses. This streaming edge coloring problem has been studied by several works in recent years. When the input graph contains $n$ vertices and has maximum vertex degree $\Delta$, it is known that in the W-streaming model, an $O(\Delta^2)$-edge coloring can be computed deterministically with $\tilde{O}(n)$ space [Ansari, Saneian, and Zarrabi-Zadeh, 2022], or an $O(\Delta^{1.5})$-edge coloring can be computed by a $\tilde{O}(n)$-space randomized algorithm [Behnezhad, Saneian, 2024] [Chechik, Mukhtar, Zhang, 2024]. In this paper, we achieve polynomial improvement over previous results. Specifically, we show how to improve the number of colors to $\tilde{O}(\Delta^{4/3+\epsilon})$ using space $\tilde{O}(n)$ deterministically, for any constant $\epsilon > 0$. This is the first deterministic result that bypasses the quadratic bound on the number of colors while using near-linear space.

Authors: Shiri Chechik, Hongyi Chen, Tianyi Zhang

Given a graph, an edge coloring assigns colors to edges so that no pairs of adjacent edges share the same color. We are interested in edge coloring algorithms under the W-streaming model. In this model, the algorithm does not have enough memory to hold the entire graph, so the edges of the input graph are read from a data stream one by one in an unknown order, and the algorithm needs to print a valid edge coloring in an output stream. The performance of the algorithm is measured by the amount of space and the number of different colors it uses. This streaming edge coloring problem has been studied by several works in recent years. When the input graph contains $n$ vertices and has maximum vertex degree $\Delta$, it is known that in the W-streaming model, an $O(\Delta^2)$-edge coloring can be computed deterministically with $\tilde{O}(n)$ space [Ansari, Saneian, and Zarrabi-Zadeh, 2022], or an $O(\Delta^{1.5})$-edge coloring can be computed by a $\tilde{O}(n)$-space randomized algorithm [Behnezhad, Saneian, 2024] [Chechik, Mukhtar, Zhang, 2024]. In this paper, we achieve polynomial improvement over previous results. Specifically, we show how to improve the number of colors to $\tilde{O}(\Delta^{4/3+\epsilon})$ using space $\tilde{O}(n)$ deterministically, for any constant $\epsilon > 0$. This is the first deterministic result that bypasses the quadratic bound on the number of colors while using near-linear space.

Multiplicative Spanners in Minor-Free Graphs

from arXiv: Data Structures and Algorithms

Authors: Greg Bodwin, Gary Hoppenworth, Zihan Tan

In FOCS 2017, Borradaille, Le, and Wulff-Nilsen addressed a long-standing open problem by proving that minor-free graphs have light spanners. Specifically, they proved that every $K_h$-minor-free graph has a $(1+\epsilon)$-spanner of lightness $O_{\epsilon}(h \sqrt{\log h})$, hence constant when $h$ and $\epsilon$ are regarded as constants. We extend this result by showing that a more expressive size/stretch tradeoff is available. Specifically: for any positive integer $k$, every $n$-node, $K_h$-minor-free graph has a $(2k-1)$-spanner with sparsity \[O\left(h^{\frac{2}{k+1}} \cdot \text{polylog } h\right),\] and a $(1+\epsilon)(2k-1)$-spanner with lightness \[O_{\epsilon}\left(h^{\frac{2}{k+1}} \cdot \text{polylog } h \right).\] We further prove that this exponent $\frac{2}{k+1}$ is best possible, assuming the girth conjecture. At a technical level, our proofs leverage the recent improvements by Postle (2020) to the remarkable density increment theorem for minor-free graphs.

Authors: Greg Bodwin, Gary Hoppenworth, Zihan Tan

In FOCS 2017, Borradaille, Le, and Wulff-Nilsen addressed a long-standing open problem by proving that minor-free graphs have light spanners. Specifically, they proved that every $K_h$-minor-free graph has a $(1+\epsilon)$-spanner of lightness $O_{\epsilon}(h \sqrt{\log h})$, hence constant when $h$ and $\epsilon$ are regarded as constants. We extend this result by showing that a more expressive size/stretch tradeoff is available. Specifically: for any positive integer $k$, every $n$-node, $K_h$-minor-free graph has a $(2k-1)$-spanner with sparsity \[O\left(h^{\frac{2}{k+1}} \cdot \text{polylog } h\right),\] and a $(1+\epsilon)(2k-1)$-spanner with lightness \[O_{\epsilon}\left(h^{\frac{2}{k+1}} \cdot \text{polylog } h \right).\] We further prove that this exponent $\frac{2}{k+1}$ is best possible, assuming the girth conjecture. At a technical level, our proofs leverage the recent improvements by Postle (2020) to the remarkable density increment theorem for minor-free graphs.

An Explicit and Efficient $O(n^2)$-Time Algorithm for Sorting Sumsets

from arXiv: Data Structures and Algorithms

Authors: S. Mundhra

We present the first explicit comparison-based algorithm that sorts the sumset $X + Y = \{x_i + y_j,\ \forall 0 \le i, j < n\}$, where $X$ and $Y$ are sorted arrays of real numbers, in optimal $O(n^2)$ time and comparisons. While Fredman (1976) proved the theoretical existence of such an algorithm, a concrete construction has remained open for nearly five decades. Our algorithm exploits the structured monotonicity of the sumset matrix to perform amortized constant-comparisons and insertions, eliminating the $\log(n)$ overhead typical of comparison-based sorting. We prove correctness and optimality in the standard comparison model, extend the method to $k$-fold sumsets with $O(n^k)$ performance, and outline potential support for dynamic updates. Experimental benchmarks show significant speedups over classical algorithms such as MergeSort and QuickSort when applied to sumsets. These results resolve a longstanding open problem in sorting theory and contribute novel techniques for exploiting input structure in algorithm design.

Authors: S. Mundhra

We present the first explicit comparison-based algorithm that sorts the sumset $X + Y = \{x_i + y_j,\ \forall 0 \le i, j < n\}$, where $X$ and $Y$ are sorted arrays of real numbers, in optimal $O(n^2)$ time and comparisons. While Fredman (1976) proved the theoretical existence of such an algorithm, a concrete construction has remained open for nearly five decades. Our algorithm exploits the structured monotonicity of the sumset matrix to perform amortized constant-comparisons and insertions, eliminating the $\log(n)$ overhead typical of comparison-based sorting. We prove correctness and optimality in the standard comparison model, extend the method to $k$-fold sumsets with $O(n^k)$ performance, and outline potential support for dynamic updates. Experimental benchmarks show significant speedups over classical algorithms such as MergeSort and QuickSort when applied to sumsets. These results resolve a longstanding open problem in sorting theory and contribute novel techniques for exploiting input structure in algorithm design.

Fully Scalable MPC Algorithms for Euclidean k-Center

from arXiv: Data Structures and Algorithms

Authors: Artur Czumaj, Guichen Gao, Mohsen Ghaffari, Shaofeng H. -C. Jiang

The $k$-center problem is a fundamental optimization problem with numerous applications in machine learning, data analysis, data mining, and communication networks. The $k$-center problem has been extensively studied in the classical sequential setting for several decades, and more recently there have been some efforts in understanding the problem in parallel computing, on the Massively Parallel Computation (MPC) model. For now, we have a good understanding of $k$-center in the case where each local MPC machine has sufficient local memory to store some representatives from each cluster, that is, when one has $\Omega(k)$ local memory per machine. While this setting covers the case of small values of $k$, for a large number of clusters these algorithms require undesirably large local memory, making them poorly scalable. The case of large $k$ has been considered only recently for the fully scalable low-local-memory MPC model for the Euclidean instances of the $k$-center problem. However, the earlier works have been considering only the constant dimensional Euclidean space, required a super-constant number of rounds, and produced only $k(1+o(1))$ centers whose cost is a super-constant approximation of $k$-center. In this work, we significantly improve upon the earlier results for the $k$-center problem for the fully scalable low-local-memory MPC model. In the low dimensional Euclidean case in $\mathbb{R}^d$, we present the first constant-round fully scalable MPC algorithm for $(2+\varepsilon)$-approximation. We push the ratio further to $(1 + \varepsilon)$-approximation albeit using slightly more $(1 + \varepsilon)k$ centers. All these results naturally extends to slightly super-constant values of $d$. In the high-dimensional regime, we provide the first fully scalable MPC algorithm that in a constant number of rounds achieves an $O(\log n/ \log \log n)$-approximation for $k$-center.

Authors: Artur Czumaj, Guichen Gao, Mohsen Ghaffari, Shaofeng H. -C. Jiang

The $k$-center problem is a fundamental optimization problem with numerous applications in machine learning, data analysis, data mining, and communication networks. The $k$-center problem has been extensively studied in the classical sequential setting for several decades, and more recently there have been some efforts in understanding the problem in parallel computing, on the Massively Parallel Computation (MPC) model. For now, we have a good understanding of $k$-center in the case where each local MPC machine has sufficient local memory to store some representatives from each cluster, that is, when one has $\Omega(k)$ local memory per machine. While this setting covers the case of small values of $k$, for a large number of clusters these algorithms require undesirably large local memory, making them poorly scalable. The case of large $k$ has been considered only recently for the fully scalable low-local-memory MPC model for the Euclidean instances of the $k$-center problem. However, the earlier works have been considering only the constant dimensional Euclidean space, required a super-constant number of rounds, and produced only $k(1+o(1))$ centers whose cost is a super-constant approximation of $k$-center. In this work, we significantly improve upon the earlier results for the $k$-center problem for the fully scalable low-local-memory MPC model. In the low dimensional Euclidean case in $\mathbb{R}^d$, we present the first constant-round fully scalable MPC algorithm for $(2+\varepsilon)$-approximation. We push the ratio further to $(1 + \varepsilon)$-approximation albeit using slightly more $(1 + \varepsilon)k$ centers. All these results naturally extends to slightly super-constant values of $d$. In the high-dimensional regime, we provide the first fully scalable MPC algorithm that in a constant number of rounds achieves an $O(\log n/ \log \log n)$-approximation for $k$-center.

Universal Online Contention Resolution with Preselected Order

from arXiv: Data Structures and Algorithms

Authors: Junyao Zhao

Online contention resolution scheme (OCRS) is a powerful technique for online decision making, which--in the case of matroids--given a matroid and a prior distribution of active elements, selects a subset of active elements that satisfies the matroid constraint in an online fashion. OCRS has been studied mostly for product distributions in the literature. Recently, universal OCRS, that works even for correlated distributions, has gained interest, because it naturally generalizes the classic notion, and its existence in the random-order arrival model turns out to be equivalent to the matroid secretary conjecture. However, currently very little is known about how to design universal OCRSs for any arrival model. In this work, we consider a natural and relatively flexible arrival model, where the OCRS is allowed to preselect (i.e., non-adaptively select) the arrival order of the elements, and within this model, we design simple and optimal universal OCRSs that are computationally efficient. In the course of deriving our OCRSs, we also discover an efficient reduction from universal online contention resolution to the matroid secretary problem for any arrival model, answering a question from Dughmi (2020).

Authors: Junyao Zhao

Online contention resolution scheme (OCRS) is a powerful technique for online decision making, which--in the case of matroids--given a matroid and a prior distribution of active elements, selects a subset of active elements that satisfies the matroid constraint in an online fashion. OCRS has been studied mostly for product distributions in the literature. Recently, universal OCRS, that works even for correlated distributions, has gained interest, because it naturally generalizes the classic notion, and its existence in the random-order arrival model turns out to be equivalent to the matroid secretary conjecture. However, currently very little is known about how to design universal OCRSs for any arrival model. In this work, we consider a natural and relatively flexible arrival model, where the OCRS is allowed to preselect (i.e., non-adaptively select) the arrival order of the elements, and within this model, we design simple and optimal universal OCRSs that are computationally efficient. In the course of deriving our OCRSs, we also discover an efficient reduction from universal online contention resolution to the matroid secretary problem for any arrival model, answering a question from Dughmi (2020).

Near-optimal Hypergraph Sparsification in Insertion-only and Bounded-deletion Streams

from arXiv: Data Structures and Algorithms

Authors: Sanjeev Khanna, Aaron Putterman, Madhu Sudan

We study the problem of constructing hypergraph cut sparsifiers in the streaming model where a hypergraph on $n$ vertices is revealed either via an arbitrary sequence of hyperedge insertions alone ({\em insertion-only} streaming model) or via an arbitrary sequence of hyperedge insertions and deletions ({\em dynamic} streaming model). For any $\epsilon \in (0,1)$, a $(1 \pm \epsilon)$ hypergraph cut-sparsifier of a hypergraph $H$ is a reweighted subgraph $H'$ whose cut values approximate those of $H$ to within a $(1 \pm \epsilon)$ factor. Prior work shows that in the static setting, one can construct a $(1 \pm \epsilon)$ hypergraph cut-sparsifier using $\tilde{O}(nr/\epsilon^2)$ bits of space [Chen-Khanna-Nagda FOCS 2020], and in the setting of dynamic streams using $\tilde{O}(nr\log m/\epsilon^2)$ bits of space [Khanna-Putterman-Sudan FOCS 2024]; here the $\tilde{O}$ notation hides terms that are polylogarithmic in $n$, and we use $m$ to denote the total number of hyperedges in the hypergraph. Up until now, the best known space complexity for insertion-only streams has been the same as that for the dynamic streams. This naturally poses the question of understanding the complexity of hypergraph sparsification in insertion-only streams. Perhaps surprisingly, in this work we show that in \emph{insertion-only} streams, a $(1 \pm \epsilon)$ cut-sparsifier can be computed in $\tilde{O}(nr/\epsilon^2)$ bits of space, \emph{matching the complexity} of the static setting. As a consequence, this also establishes an $\Omega(\log m)$ factor separation between the space complexity of hypergraph cut sparsification in insertion-only streams and dynamic streams, as the latter is provably known to require $\Omega(nr \log m)$ bits of space.

Authors: Sanjeev Khanna, Aaron Putterman, Madhu Sudan

We study the problem of constructing hypergraph cut sparsifiers in the streaming model where a hypergraph on $n$ vertices is revealed either via an arbitrary sequence of hyperedge insertions alone ({\em insertion-only} streaming model) or via an arbitrary sequence of hyperedge insertions and deletions ({\em dynamic} streaming model). For any $\epsilon \in (0,1)$, a $(1 \pm \epsilon)$ hypergraph cut-sparsifier of a hypergraph $H$ is a reweighted subgraph $H'$ whose cut values approximate those of $H$ to within a $(1 \pm \epsilon)$ factor. Prior work shows that in the static setting, one can construct a $(1 \pm \epsilon)$ hypergraph cut-sparsifier using $\tilde{O}(nr/\epsilon^2)$ bits of space [Chen-Khanna-Nagda FOCS 2020], and in the setting of dynamic streams using $\tilde{O}(nr\log m/\epsilon^2)$ bits of space [Khanna-Putterman-Sudan FOCS 2024]; here the $\tilde{O}$ notation hides terms that are polylogarithmic in $n$, and we use $m$ to denote the total number of hyperedges in the hypergraph. Up until now, the best known space complexity for insertion-only streams has been the same as that for the dynamic streams. This naturally poses the question of understanding the complexity of hypergraph sparsification in insertion-only streams. Perhaps surprisingly, in this work we show that in \emph{insertion-only} streams, a $(1 \pm \epsilon)$ cut-sparsifier can be computed in $\tilde{O}(nr/\epsilon^2)$ bits of space, \emph{matching the complexity} of the static setting. As a consequence, this also establishes an $\Omega(\log m)$ factor separation between the space complexity of hypergraph cut sparsification in insertion-only streams and dynamic streams, as the latter is provably known to require $\Omega(nr \log m)$ bits of space.

Linear Time Subsequence and Supersequence Regex Matching

from arXiv: Data Structures and Algorithms

Authors: Antoine Amarilli, Florin Manea, Tina Ringleb, Markus L. Schmid

It is well-known that checking whether a given string $w$ matches a given regular expression $r$ can be done in quadratic time $O(|w|\cdot |r|)$ and that this cannot be improved to a truly subquadratic running time of $O((|w|\cdot |r|)^{1-\epsilon})$ assuming the strong exponential time hypothesis (SETH). We study a different matching paradigm where we ask instead whether $w$ has a subsequence that matches $r$, and show that regex matching in this sense can be solved in linear time $O(|w| + |r|)$. Further, the same holds if we ask for a supersequence. We show that the quantitative variants where we want to compute a longest or shortest subsequence or supersequence of $w$ that matches $r$ can be solved in $O(|w| \cdot |r|)$, i. e., asymptotically no worse than classical regex matching; and we show that $O(|w| + |r|)$ is conditionally not possible for these problems. We also investigate these questions with respect to other natural string relations like the infix, prefix, left-extension or extension relation instead of the subsequence and supersequence relation. We further study the complexity of the universal problem where we ask if all subsequences (or supersequences, infixes, prefixes, left-extensions or extensions) of an input string satisfy a given regular expression.

Authors: Antoine Amarilli, Florin Manea, Tina Ringleb, Markus L. Schmid

It is well-known that checking whether a given string $w$ matches a given regular expression $r$ can be done in quadratic time $O(|w|\cdot |r|)$ and that this cannot be improved to a truly subquadratic running time of $O((|w|\cdot |r|)^{1-\epsilon})$ assuming the strong exponential time hypothesis (SETH). We study a different matching paradigm where we ask instead whether $w$ has a subsequence that matches $r$, and show that regex matching in this sense can be solved in linear time $O(|w| + |r|)$. Further, the same holds if we ask for a supersequence. We show that the quantitative variants where we want to compute a longest or shortest subsequence or supersequence of $w$ that matches $r$ can be solved in $O(|w| \cdot |r|)$, i. e., asymptotically no worse than classical regex matching; and we show that $O(|w| + |r|)$ is conditionally not possible for these problems. We also investigate these questions with respect to other natural string relations like the infix, prefix, left-extension or extension relation instead of the subsequence and supersequence relation. We further study the complexity of the universal problem where we ask if all subsequences (or supersequences, infixes, prefixes, left-extensions or extensions) of an input string satisfy a given regular expression.

Fast, Space-Optimal Streaming Algorithms for Clustering and Subspace Embeddings

from arXiv: Data Structures and Algorithms

Authors: Vincent Cohen-Addad, Liudeng Wang, David P. Woodruff, Samson Zhou

We show that both clustering and subspace embeddings can be performed in the streaming model with the same asymptotic efficiency as in the central/offline setting. For $(k, z)$-clustering in the streaming model, we achieve a number of words of memory which is independent of the number $n$ of input points and the aspect ratio $\Delta$, yielding an optimal bound of $\tilde{\mathcal{O}}\left(\frac{dk}{\min(\varepsilon^4,\varepsilon^{z+2})}\right)$ words for accuracy parameter $\varepsilon$ on $d$-dimensional points. Additionally, we obtain amortized update time of $d\,\log(k)\cdot\text{polylog}(\log(n\Delta))$, which is an exponential improvement over the previous $d\,\text{poly}(k,\log(n\Delta))$. Our method also gives the fastest runtime for $(k,z)$-clustering even in the offline setting. For subspace embeddings in the streaming model, we achieve $\mathcal{O}(d)$ update time and space-optimal constructions, using $\tilde{\mathcal{O}}\left(\frac{d^2}{\varepsilon^2}\right)$ words for $p\le 2$ and $\tilde{\mathcal{O}}\left(\frac{d^{p/2+1}}{\varepsilon^2}\right)$ words for $p>2$, showing that streaming algorithms can match offline algorithms in both space and time complexity.

Authors: Vincent Cohen-Addad, Liudeng Wang, David P. Woodruff, Samson Zhou

We show that both clustering and subspace embeddings can be performed in the streaming model with the same asymptotic efficiency as in the central/offline setting. For $(k, z)$-clustering in the streaming model, we achieve a number of words of memory which is independent of the number $n$ of input points and the aspect ratio $\Delta$, yielding an optimal bound of $\tilde{\mathcal{O}}\left(\frac{dk}{\min(\varepsilon^4,\varepsilon^{z+2})}\right)$ words for accuracy parameter $\varepsilon$ on $d$-dimensional points. Additionally, we obtain amortized update time of $d\,\log(k)\cdot\text{polylog}(\log(n\Delta))$, which is an exponential improvement over the previous $d\,\text{poly}(k,\log(n\Delta))$. Our method also gives the fastest runtime for $(k,z)$-clustering even in the offline setting. For subspace embeddings in the streaming model, we achieve $\mathcal{O}(d)$ update time and space-optimal constructions, using $\tilde{\mathcal{O}}\left(\frac{d^2}{\varepsilon^2}\right)$ words for $p\le 2$ and $\tilde{\mathcal{O}}\left(\frac{d^{p/2+1}}{\varepsilon^2}\right)$ words for $p>2$, showing that streaming algorithms can match offline algorithms in both space and time complexity.

A Theory of Spectral CSP Sparsification

from arXiv: Data Structures and Algorithms

Authors: Sanjeev Khanna, Aaron Putterman, Madhu Sudan

We initiate the study of spectral sparsification for instances of Constraint Satisfaction Problems (CSPs). In particular, we introduce a notion of the \emph{spectral energy} of a fractional assignment for a Boolean CSP instance, and define a \emph{spectral sparsifier} as a weighted subset of constraints that approximately preserves this energy for all fractional assignments. Our definition not only strengthens the combinatorial notion of a CSP sparsifier but also extends well-studied concepts such as spectral sparsifiers for graphs and hypergraphs. Recent work by Khanna, Putterman, and Sudan [SODA 2024] demonstrated near-linear sized \emph{combinatorial sparsifiers} for a broad class of CSPs, which they term \emph{field-affine CSPs}. Our main result is a polynomial-time algorithm that constructs a spectral CSP sparsifier of near-quadratic size for all field-affine CSPs. This class of CSPs includes graph (and hypergraph) cuts, XORs, and more generally, any predicate which can be written as $P(x_1, \dots x_r) = \mathbf{1}[\sum a_i x_i \neq b \mod p]$. Based on our notion of the spectral energy of a fractional assignment, we also define an analog of the second eigenvalue of a CSP instance. We then show an extension of Cheeger's inequality for all even-arity XOR CSPs, showing that this second eigenvalue loosely captures the ``expansion'' of the underlying CSP. This extension specializes to the case of Cheeger's inequality when all constraints are even XORs and thus gives a new generalization of this powerful inequality which converts the combinatorial notion of expansion to an analytic property.

Authors: Sanjeev Khanna, Aaron Putterman, Madhu Sudan

We initiate the study of spectral sparsification for instances of Constraint Satisfaction Problems (CSPs). In particular, we introduce a notion of the \emph{spectral energy} of a fractional assignment for a Boolean CSP instance, and define a \emph{spectral sparsifier} as a weighted subset of constraints that approximately preserves this energy for all fractional assignments. Our definition not only strengthens the combinatorial notion of a CSP sparsifier but also extends well-studied concepts such as spectral sparsifiers for graphs and hypergraphs. Recent work by Khanna, Putterman, and Sudan [SODA 2024] demonstrated near-linear sized \emph{combinatorial sparsifiers} for a broad class of CSPs, which they term \emph{field-affine CSPs}. Our main result is a polynomial-time algorithm that constructs a spectral CSP sparsifier of near-quadratic size for all field-affine CSPs. This class of CSPs includes graph (and hypergraph) cuts, XORs, and more generally, any predicate which can be written as $P(x_1, \dots x_r) = \mathbf{1}[\sum a_i x_i \neq b \mod p]$. Based on our notion of the spectral energy of a fractional assignment, we also define an analog of the second eigenvalue of a CSP instance. We then show an extension of Cheeger's inequality for all even-arity XOR CSPs, showing that this second eigenvalue loosely captures the ``expansion'' of the underlying CSP. This extension specializes to the case of Cheeger's inequality when all constraints are even XORs and thus gives a new generalization of this powerful inequality which converts the combinatorial notion of expansion to an analytic property.

Wednesday, April 23

Real People

from Computational Complexity

Right after the election I wrote a post predicting what would happen to higher education under Trump, most of which is coming true, but I had a massive failure of imagination missing the direct attacks on major universities. I won't detail all the challenges to higher ed, which change daily with every new executive order and court ruling. The Chronicle has a good tracker of these changes.

But often lost in all the news are the actual people who aren't making the news hurt by these actions, through no fault of their own: A student who had his visa cancelled while out of the country so he can't get back in. PhD students who just lost their funding when their advisor's grant was cancelled. A postdoc in a similar situation just months before he starts an academic job. A young faculty member who had to hold off submitting a Career award proposal until a TRO restored the indirect cost. A recently tenured professor at a good US university interviewing outside of the country. Potential students in other countries trying to decide if they should still go to school and build a life in the US. 

The number of people, grants and universities affected is still on the low end, nearly all students continue to study, graduate and work without any problems, and many universities have offered legal and financial support to affected students and faculty. In my humble opinion, the strong educational opportunities inside the US still greatly exceed those outside. Universities have weathered many challenges before: McCarthyism in the 50s, campus occupations and protests in the 60s, and budget challenges from the great depression, to the fiscal crisis and covid. Trump's time as president has an end date and we'll get through all of this, but it requires all of us to push back and remind the administration and the public about the important role our universities play in the health of the country.

And as much as it pains me to say this as a Cornell alum, I'm rooting for Harvard.

By Lance Fortnow

Right after the election I wrote a post predicting what would happen to higher education under Trump, most of which is coming true, but I had a massive failure of imagination missing the direct attacks on major universities. I won't detail all the challenges to higher ed, which change daily with every new executive order and court ruling. The Chronicle has a good tracker of these changes.

But often lost in all the news are the actual people who aren't making the news hurt by these actions, through no fault of their own: A student who had his visa cancelled while out of the country so he can't get back in. PhD students who just lost their funding when their advisor's grant was cancelled. A postdoc in a similar situation just months before he starts an academic job. A young faculty member who had to hold off submitting a Career award proposal until a TRO restored the indirect cost. A recently tenured professor at a good US university interviewing outside of the country. Potential students in other countries trying to decide if they should still go to school and build a life in the US. 

The number of people, grants and universities affected is still on the low end, nearly all students continue to study, graduate and work without any problems, and many universities have offered legal and financial support to affected students and faculty. In my humble opinion, the strong educational opportunities inside the US still greatly exceed those outside. Universities have weathered many challenges before: McCarthyism in the 50s, campus occupations and protests in the 60s, and budget challenges from the great depression, to the fiscal crisis and covid. Trump's time as president has an end date and we'll get through all of this, but it requires all of us to push back and remind the administration and the public about the important role our universities play in the health of the country.

And as much as it pains me to say this as a Cornell alum, I'm rooting for Harvard.

By Lance Fortnow

TR25-053 | On Approximate Symmetric Polynomials and Tightness of Homogenization Results | Amir Shpilka

from ECCC Papers

Motivated by questions concerning the multilinear and homogeneous complexity of the elementary symmetric polynomials, we prove the following results: We first show that by making small modifications to the nonzero coefficients of the degree-$K$, $N$-variate elementary symmetric polynomial $\sigma_{N,K}$, one obtains a polynomial that can be computed by a monotone formula of size $K^{O(\log K)} \cdot N$. As a corollary, we show that a result of Raz [Raz13] concerning the homogenization of algebraic multilinear or monotone formulas is tight. Another corollary is that the monotone bounded rigidity of the inclusion matrix between $K$-subsets and $N-K$ subsets of a universe of size $N$ is small.

Motivated by questions concerning the multilinear and homogeneous complexity of the elementary symmetric polynomials, we prove the following results: We first show that by making small modifications to the nonzero coefficients of the degree-$K$, $N$-variate elementary symmetric polynomial $\sigma_{N,K}$, one obtains a polynomial that can be computed by a monotone formula of size $K^{O(\log K)} \cdot N$. As a corollary, we show that a result of Raz [Raz13] concerning the homogenization of algebraic multilinear or monotone formulas is tight. Another corollary is that the monotone bounded rigidity of the inclusion matrix between $K$-subsets and $N-K$ subsets of a universe of size $N$ is small.

A Mysterious Connection Between Tolerant Junta Testing and Agnostically Learning Conjunctions

from arXiv: Data Structures and Algorithms

Authors: Xi Chen, Shyamal Patel, Rocco A. Servedio

The main conceptual contribution of this paper is identifying a previously unnoticed connection between two central problems in computational learning theory and property testing: agnostically learning conjunctions and tolerantly testing juntas. Inspired by this connection, the main technical contribution is a pair of improved algorithms for these two problems. In more detail, - We give a distribution-free algorithm for agnostically PAC learning conjunctions over $\{\pm 1\}^n$ that runs in time $2^{\widetilde{O}(n^{1/3})}$, for constant excess error $\varepsilon$. This improves on the fastest previously published algorithm, which runs in time $2^{\widetilde{O}(n^{1/2})}$ [KKMS08]. - Building on the ideas in our agnostic conjunction learner and using significant additional technical ingredients, we give an adaptive tolerant testing algorithm for $k$-juntas that makes $2^{\widetilde{O}(k^{1/3})}$ queries, for constant "gap parameter" $\varepsilon$ between the "near" and "far" cases. This improves on the best previous results, due to [ITW21, NP24], which make $2^{\widetilde{O}(\sqrt{k})}$ queries. Since there is a known $2^{\widetilde{\Omega}(\sqrt{k})}$ lower bound for non-adaptive tolerant junta testers, our result shows that adaptive tolerant junta testing algorithms provably outperform non-adaptive ones.

Authors: Xi Chen, Shyamal Patel, Rocco A. Servedio

The main conceptual contribution of this paper is identifying a previously unnoticed connection between two central problems in computational learning theory and property testing: agnostically learning conjunctions and tolerantly testing juntas. Inspired by this connection, the main technical contribution is a pair of improved algorithms for these two problems. In more detail, - We give a distribution-free algorithm for agnostically PAC learning conjunctions over $\{\pm 1\}^n$ that runs in time $2^{\widetilde{O}(n^{1/3})}$, for constant excess error $\varepsilon$. This improves on the fastest previously published algorithm, which runs in time $2^{\widetilde{O}(n^{1/2})}$ [KKMS08]. - Building on the ideas in our agnostic conjunction learner and using significant additional technical ingredients, we give an adaptive tolerant testing algorithm for $k$-juntas that makes $2^{\widetilde{O}(k^{1/3})}$ queries, for constant "gap parameter" $\varepsilon$ between the "near" and "far" cases. This improves on the best previous results, due to [ITW21, NP24], which make $2^{\widetilde{O}(\sqrt{k})}$ queries. Since there is a known $2^{\widetilde{\Omega}(\sqrt{k})}$ lower bound for non-adaptive tolerant junta testers, our result shows that adaptive tolerant junta testing algorithms provably outperform non-adaptive ones.

Branch-and-Bound Algorithms as Polynomial-time Approximation Schemes

from arXiv: Data Structures and Algorithms

Authors: Koppány István Encz, Monaldo Mastrolilli, Eleonora Vercesi

Branch-and-bound algorithms (B&B) and polynomial-time approximation schemes (PTAS) are two seemingly distant areas of combinatorial optimization. We intend to (partially) bridge the gap between them while expanding the boundary of theoretical knowledge on the B&B framework. Branch-and-bound algorithms typically guarantee that an optimal solution is eventually found. However, we show that the standard implementation of branch-and-bound for certain knapsack and scheduling problems also exhibits PTAS-like behavior, yielding increasingly better solutions within polynomial time. Our findings are supported by computational experiments and comparisons with benchmark methods. This paper is an extended version of a paper accepted at ICALP 2025.

Authors: Koppány István Encz, Monaldo Mastrolilli, Eleonora Vercesi

Branch-and-bound algorithms (B&B) and polynomial-time approximation schemes (PTAS) are two seemingly distant areas of combinatorial optimization. We intend to (partially) bridge the gap between them while expanding the boundary of theoretical knowledge on the B&B framework. Branch-and-bound algorithms typically guarantee that an optimal solution is eventually found. However, we show that the standard implementation of branch-and-bound for certain knapsack and scheduling problems also exhibits PTAS-like behavior, yielding increasingly better solutions within polynomial time. Our findings are supported by computational experiments and comparisons with benchmark methods. This paper is an extended version of a paper accepted at ICALP 2025.

Towards True Work-Efficiency in Parallel Derandomization: MIS, Maximal Matching, and Hitting Set

from arXiv: Data Structures and Algorithms

Authors: Mohsen Ghaffari, Christoph Grunau

Derandomization is one of the classic topics studied in the theory of parallel computations, dating back to the early 1980s. Despite much work, all known techniques lead to deterministic algorithms that are not work-efficient. For instance, for the well-studied problem of maximal independent set -- e.g., [Karp, Wigderson STOC'84; Luby STOC' 85; Luby FOCS'88] -- state-of-the-art deterministic algorithms require at least $m \cdot poly(\log n)$ work, where $m$ and $n$ denote the number of edges and vertices. Hence, these deterministic algorithms will remain slower than their trivial sequential counterparts unless we have at least $poly(\log n)$ processors. In this paper, we present a generic parallel derandomization technique that moves exponentially closer to work-efficiency. The method iteratively rounds fractional solutions representing the randomized assignments to integral solutions that provide deterministic assignments, while maintaining certain linear or quadratic objective functions, and in an \textit{essentially work-efficient} manner. As example end-results, we use this technique to obtain deterministic algorithms with $m \cdot poly(\log \log n)$ work and $poly(\log n)$ depth for problems such as maximal independent set, maximal matching, and hitting set.

Authors: Mohsen Ghaffari, Christoph Grunau

Derandomization is one of the classic topics studied in the theory of parallel computations, dating back to the early 1980s. Despite much work, all known techniques lead to deterministic algorithms that are not work-efficient. For instance, for the well-studied problem of maximal independent set -- e.g., [Karp, Wigderson STOC'84; Luby STOC' 85; Luby FOCS'88] -- state-of-the-art deterministic algorithms require at least $m \cdot poly(\log n)$ work, where $m$ and $n$ denote the number of edges and vertices. Hence, these deterministic algorithms will remain slower than their trivial sequential counterparts unless we have at least $poly(\log n)$ processors. In this paper, we present a generic parallel derandomization technique that moves exponentially closer to work-efficiency. The method iteratively rounds fractional solutions representing the randomized assignments to integral solutions that provide deterministic assignments, while maintaining certain linear or quadratic objective functions, and in an \textit{essentially work-efficient} manner. As example end-results, we use this technique to obtain deterministic algorithms with $m \cdot poly(\log \log n)$ work and $poly(\log n)$ depth for problems such as maximal independent set, maximal matching, and hitting set.

Quantum Speedup for Sampling Random Spanning Trees

from arXiv: Data Structures and Algorithms

Authors: Chenghua Liu, Minbo Gao, Zhengfeng Ji, Simon Apers

We present a quantum algorithm for sampling random spanning trees from a weighted graph in $\widetilde{O}(\sqrt{mn})$ time, where $n$ and $m$ denote the number of vertices and edges, respectively. Our algorithm has sublinear runtime for dense graphs and achieves a quantum speedup over the best-known classical algorithm, which runs in $\widetilde{O}(m)$ time. The approach carefully combines, on one hand, a classical method based on ``large-step'' random walks for reduced mixing time and, on the other hand, quantum algorithmic techniques, including quantum graph sparsification and a sampling-without-replacement variant of Hamoudi's multiple-state preparation. We also establish a matching lower bound, proving the optimality of our algorithm up to polylogarithmic factors. These results highlight the potential of quantum computing in accelerating fundamental graph sampling problems.

Authors: Chenghua Liu, Minbo Gao, Zhengfeng Ji, Simon Apers

We present a quantum algorithm for sampling random spanning trees from a weighted graph in $\widetilde{O}(\sqrt{mn})$ time, where $n$ and $m$ denote the number of vertices and edges, respectively. Our algorithm has sublinear runtime for dense graphs and achieves a quantum speedup over the best-known classical algorithm, which runs in $\widetilde{O}(m)$ time. The approach carefully combines, on one hand, a classical method based on ``large-step'' random walks for reduced mixing time and, on the other hand, quantum algorithmic techniques, including quantum graph sparsification and a sampling-without-replacement variant of Hamoudi's multiple-state preparation. We also establish a matching lower bound, proving the optimality of our algorithm up to polylogarithmic factors. These results highlight the potential of quantum computing in accelerating fundamental graph sampling problems.

Smooth Calibration and Decision Making

from arXiv: Data Structures and Algorithms

Authors: Jason Hartline, Yifan Wu, Yunran Yang

Calibration requires predictor outputs to be consistent with their Bayesian posteriors. For machine learning predictors that do not distinguish between small perturbations, calibration errors are continuous in predictions, e.g., smooth calibration error (Foster and Hart, 2018), Distance to Calibration (Blasiok et al., 2023a). On the contrary, decision-makers who use predictions make optimal decisions discontinuously in probabilistic space, experiencing loss from miscalibration discontinuously. Calibration errors for decision-making are thus discontinuous, e.g., Expected Calibration Error (Foster and Vohra, 1997), and Calibration Decision Loss (Hu and Wu, 2024). Thus, predictors with a low calibration error for machine learning may suffer a high calibration error for decision-making, i.e., they may not be trustworthy for decision-makers optimizing assuming their predictions are correct. It is natural to ask if post-processing a predictor with a low calibration error for machine learning is without loss to achieve a low calibration error for decision-making. In our paper, we show that post-processing an online predictor with $\epsilon$ distance to calibration achieves $O(\sqrt{\epsilon})$ ECE and CDL, which is asymptotically optimal. The post-processing algorithm adds noise to make predictions differentially private. The optimal bound from low distance to calibration predictors from post-processing is non-optimal compared with existing online calibration algorithms that directly optimize for ECE and CDL.

Authors: Jason Hartline, Yifan Wu, Yunran Yang

Calibration requires predictor outputs to be consistent with their Bayesian posteriors. For machine learning predictors that do not distinguish between small perturbations, calibration errors are continuous in predictions, e.g., smooth calibration error (Foster and Hart, 2018), Distance to Calibration (Blasiok et al., 2023a). On the contrary, decision-makers who use predictions make optimal decisions discontinuously in probabilistic space, experiencing loss from miscalibration discontinuously. Calibration errors for decision-making are thus discontinuous, e.g., Expected Calibration Error (Foster and Vohra, 1997), and Calibration Decision Loss (Hu and Wu, 2024). Thus, predictors with a low calibration error for machine learning may suffer a high calibration error for decision-making, i.e., they may not be trustworthy for decision-makers optimizing assuming their predictions are correct. It is natural to ask if post-processing a predictor with a low calibration error for machine learning is without loss to achieve a low calibration error for decision-making. In our paper, we show that post-processing an online predictor with $\epsilon$ distance to calibration achieves $O(\sqrt{\epsilon})$ ECE and CDL, which is asymptotically optimal. The post-processing algorithm adds noise to make predictions differentially private. The optimal bound from low distance to calibration predictors from post-processing is non-optimal compared with existing online calibration algorithms that directly optimize for ECE and CDL.

On the Price of Differential Privacy for Hierarchical Clustering

from arXiv: Data Structures and Algorithms

Authors: Chengyuan Deng, Jie Gao, Jalaj Upadhyay, Chen Wang, Samson Zhou

Hierarchical clustering is a fundamental unsupervised machine learning task with the aim of organizing data into a hierarchy of clusters. Many applications of hierarchical clustering involve sensitive user information, therefore motivating recent studies on differentially private hierarchical clustering under the rigorous framework of Dasgupta's objective. However, it has been shown that any privacy-preserving algorithm under edge-level differential privacy necessarily suffers a large error. To capture practical applications of this problem, we focus on the weight privacy model, where each edge of the input graph is at least unit weight. We present a novel algorithm in the weight privacy model that shows significantly better approximation than known impossibility results in the edge-level DP setting. In particular, our algorithm achieves $O(\log^{1.5}n/\varepsilon)$ multiplicative error for $\varepsilon$-DP and runs in polynomial time, where $n$ is the size of the input graph, and the cost is never worse than the optimal additive error in existing work. We complement our algorithm by showing if the unit-weight constraint does not apply, the lower bound for weight-level DP hierarchical clustering is essentially the same as the edge-level DP, i.e. $\Omega(n^2/\varepsilon)$ additive error. As a result, we also obtain a new lower bound of $\tilde{\Omega}(1/\varepsilon)$ additive error for balanced sparsest cuts in the weight-level DP model, which may be of independent interest. Finally, we evaluate our algorithm on synthetic and real-world datasets. Our experimental results show that our algorithm performs well in terms of extra cost and has good scalability to large graphs.

Authors: Chengyuan Deng, Jie Gao, Jalaj Upadhyay, Chen Wang, Samson Zhou

Hierarchical clustering is a fundamental unsupervised machine learning task with the aim of organizing data into a hierarchy of clusters. Many applications of hierarchical clustering involve sensitive user information, therefore motivating recent studies on differentially private hierarchical clustering under the rigorous framework of Dasgupta's objective. However, it has been shown that any privacy-preserving algorithm under edge-level differential privacy necessarily suffers a large error. To capture practical applications of this problem, we focus on the weight privacy model, where each edge of the input graph is at least unit weight. We present a novel algorithm in the weight privacy model that shows significantly better approximation than known impossibility results in the edge-level DP setting. In particular, our algorithm achieves $O(\log^{1.5}n/\varepsilon)$ multiplicative error for $\varepsilon$-DP and runs in polynomial time, where $n$ is the size of the input graph, and the cost is never worse than the optimal additive error in existing work. We complement our algorithm by showing if the unit-weight constraint does not apply, the lower bound for weight-level DP hierarchical clustering is essentially the same as the edge-level DP, i.e. $\Omega(n^2/\varepsilon)$ additive error. As a result, we also obtain a new lower bound of $\tilde{\Omega}(1/\varepsilon)$ additive error for balanced sparsest cuts in the weight-level DP model, which may be of independent interest. Finally, we evaluate our algorithm on synthetic and real-world datasets. Our experimental results show that our algorithm performs well in terms of extra cost and has good scalability to large graphs.

Adaptivity Gaps for Stochastic Probing with Subadditive Functions

from arXiv: Data Structures and Algorithms

Authors: Jian Li, Yinchen Liu, Yiran Zhang

In this paper, we study the stochastic probing problem under a general monotone norm objective. Given a ground set $U = [n]$, each element $i \in U$ has an independent nonnegative random variable $X_i$ with known distribution. Probing an element reveals its value, and the sequence of probed elements must satisfy a prefix-closed feasibility constraint $\mathcal{F}$. A monotone norm $f: \mathbb{R}_{\geq 0}^n \to \mathbb{R}_{\geq 0}$ determines the reward $f(X_P)$, where $P$ is the set of probed elements and $X_P$ is the vector with $X_i$ for $i \in P$ and 0 otherwise. The goal is to design a probing strategy maximizing the expected reward $\mathbb{E}[f(X_P)]$. We focus on the adaptivity gap: the ratio between the expected rewards of optimal adaptive and optimal non-adaptive strategies. We resolve an open question posed in [GNS17, KMS24], showing that for general monotone norms, the adaptivity gap is $O(\log^2 n)$. A refined analysis yields an improved bound of $O(\log r \log n / \log\log n)$, where $r$ is the maximum size of a feasible probing sequence. As a by-product, we derive an asymptotically tight adaptivity gap $\Theta(\log n / \log\log n)$ for Bernoulli probing with binary-XOS objectives, matching the known lower bound. Additionally, we show an $O(\log^3 n)$ upper bound for Bernoulli probing with general subadditive objectives. For monotone symmetric norms, we prove the adaptivity gap is $O(1)$, improving the previous $O(\log n)$ bound from [PRS23].

Authors: Jian Li, Yinchen Liu, Yiran Zhang

In this paper, we study the stochastic probing problem under a general monotone norm objective. Given a ground set $U = [n]$, each element $i \in U$ has an independent nonnegative random variable $X_i$ with known distribution. Probing an element reveals its value, and the sequence of probed elements must satisfy a prefix-closed feasibility constraint $\mathcal{F}$. A monotone norm $f: \mathbb{R}_{\geq 0}^n \to \mathbb{R}_{\geq 0}$ determines the reward $f(X_P)$, where $P$ is the set of probed elements and $X_P$ is the vector with $X_i$ for $i \in P$ and 0 otherwise. The goal is to design a probing strategy maximizing the expected reward $\mathbb{E}[f(X_P)]$. We focus on the adaptivity gap: the ratio between the expected rewards of optimal adaptive and optimal non-adaptive strategies. We resolve an open question posed in [GNS17, KMS24], showing that for general monotone norms, the adaptivity gap is $O(\log^2 n)$. A refined analysis yields an improved bound of $O(\log r \log n / \log\log n)$, where $r$ is the maximum size of a feasible probing sequence. As a by-product, we derive an asymptotically tight adaptivity gap $\Theta(\log n / \log\log n)$ for Bernoulli probing with binary-XOS objectives, matching the known lower bound. Additionally, we show an $O(\log^3 n)$ upper bound for Bernoulli probing with general subadditive objectives. For monotone symmetric norms, we prove the adaptivity gap is $O(1)$, improving the previous $O(\log n)$ bound from [PRS23].

Prize-Collecting Forest with Submodular Penalties: Improved Approximation

from arXiv: Data Structures and Algorithms

Authors: Ali Ahmadi, Iman Gholami, MohammadTaghi Hajiaghayi, Peyman Jabbarzade, Mohammad Mahdavi

Constrained forest problems form a class of graph problems where specific connectivity requirements for certain cuts within the graph must be satisfied by selecting the minimum-cost set of edges. The prize-collecting version of these problems introduces flexibility by allowing penalties to be paid to ignore some connectivity requirements. Goemans and Williamson introduced a general technique and developed a 2-approximation algorithm for constrained forest problems. Further, Sharma, Swamy, and Williamson extended this work by developing a 2.54-approximation algorithm for the prize-collecting version of these problems. Motivated by the generality of their framework, which includes problems such as Steiner trees, Steiner forests, and their variants, we pursued further exploration. We present a significant improvement by achieving a 2-approximation algorithm for this general model, matching the approximation factor of the constrained forest problems.

Authors: Ali Ahmadi, Iman Gholami, MohammadTaghi Hajiaghayi, Peyman Jabbarzade, Mohammad Mahdavi

Constrained forest problems form a class of graph problems where specific connectivity requirements for certain cuts within the graph must be satisfied by selecting the minimum-cost set of edges. The prize-collecting version of these problems introduces flexibility by allowing penalties to be paid to ignore some connectivity requirements. Goemans and Williamson introduced a general technique and developed a 2-approximation algorithm for constrained forest problems. Further, Sharma, Swamy, and Williamson extended this work by developing a 2.54-approximation algorithm for the prize-collecting version of these problems. Motivated by the generality of their framework, which includes problems such as Steiner trees, Steiner forests, and their variants, we pursued further exploration. We present a significant improvement by achieving a 2-approximation algorithm for this general model, matching the approximation factor of the constrained forest problems.

Tuesday, April 22

Rossi's Metallic Rules

from Ben Recht

Why do evaluations tend to find that social programs don't work?

As we head into the final week of our course on AI Evaluation, we’re reading Peter Rossi’s 1987 paper on social program evaluation, “The Iron Law of Evaluation and Other Metallic Rules.” Rossi is a fitting cap on the semester because Deb and I adopted Rossi’s definition of evaluation as the backbone of our course.1 It’s also a key reference as we begin to understand how to evaluate human-facing AI. We need to consider how to evaluate claims about the impact of AI systems on people. Do they augment or inhibit the capabilities of students, healthcare providers, or software engineers?

Rossi’s paper begins with a conundrum. He notes that rigorous program evaluation has only become a “routine” part of policy development in the last two decades (i.e., since the 1960s). However, the more evaluation becomes routine, the more it seems like nothing works. He lays out four “laws” of program evaluation that describe the state of affairs in the late 1980s. Rossi is clear that these “laws” are also socioscientific, so they don’t hold like the laws of physics. But they are a common enough state of affairs that they highlight issues in policy construction.

The laws are named after metals in decreasing order of robustness. The first law describes the general state of affairs of program evaluation: on average, nothing works.

The Iron Law of Evaluation: The expected value of any net impact assessment of any large scale social program is zero.

The second law is harsher, suggesting that the more skill, effort, and thought put into an evaluation, the less likely it is to find a benefit:

The Stainless Steel Law of Evaluation: The better designed the impact assessment of a social program, the more likely is the resulting estimate of net impact to be zero.

The third law argues that social scientific methods themselves are far less strongly influential than social scientists believe.

The Brass Law of Evaluation: The more social programs are designed to change individuals, the more likely the net impact of the program will be zero.

The final law is an optimistic assessment about selective evaluation.

The Zinc Law of Evaluation: Only those programs that are likely to fail are evaluated.

These glib soundbites still ring true. But the question remains, why do well-evaluated social programs fail to show benefit? Rossi has a peculiar answer. Most who indict the effectiveness of social scientific practice wonder if there’s an issue with the entire enterprise. Rossi, on the other hand, argues for doubling down. He claims that an improved social scientific practice will improve the effectiveness of programs. He points to several common flaws in policy implementation, where faulty understandings of social phenomena, faulty understandings of how to translate theory into intervention, or faulty implementations themselves could be corrected by better adherence to social scientific theory. More ambitiously, Rossi wants social science to build out a discipline of social engineering, drawing closer connections to practitioners in education, psychology, and public administration. He demands more rigorous and quantitative evaluation, even if that means fewer programs will be assessed as beneficial. He warns against the qualitative as having “even greater and more obvious deficiencies” than the randomized controlled trial.

Rossi closes his paper, concluding, “There are no social science equivalents of the Salk vaccine.” But he takes this as evidence of a lack of methodological discipline. Perhaps with better theory, better evaluation, and better link to practice, the field can get there.

In a set of remarks delivered in 2003 at the Association for Public Policy and Management Research Conference, Rossi in fact argues that more rigorous quantitative methods have brought social science to a better place. Without references, he asserts:

“There are quite a large number of well conducted impact assessments that yield statistically and substantively significant effect sizes. I believe that we are learning how properly to design and implement interventions that are effective.”

In these later remarks, he suggests that the reason programs continue to fail to evaluate is because of poor evaluations, not because the underlying methodology doesn’t work. If only people were more rigorous with how they did their evaluations, he thinks the laws would be refuted.

The subsequent decades have not been kind to Rossi’s predictions. Not only do social program evaluations still fail to find positive net benefit on average, but public discontent with this technocratic mindset has reached an all-time high.

While the observations of this original paper seem irrefutable, Rossi’s diagnosis was wrong. What if it wasn’t the methods holding back social science, but the entire conception of the notion of social science? Many of Rossi’s contemporaries were sounding the alarm. The 1980s are full of papers reckoning with the failures of quantitative social science. Leamer’s “Let’s Take the Con out of Econometrics.” Gigerenzer et al.’s “The Empire of Chance.” Stanley Liberson’s “Making It Count.” Meehl’s “Two Knights” paper. In the popular press, the work of Neil Postman. All of these authors came to the same conclusion: social science was something different from natural science, and hence the methods of the natural sciences, especially those of control, don’t apply to social systems and don’t align with liberal values.

Oddly, instead of finding a constructive path forward, most of social science doubled down with Rossi for forty years, accumulating more data, more methods, more computing. The reason Rossi’s paper remains a classic reading is that social programs still seem to obey the iron rules. Despite massive amounts of data, plots, and regressions, “social engineering” isn’t more effective than it was in 1987. Those Iron Laws still look pretty spot on, and it’s past time to consider that those 1980s critics of modern social science were right all along.

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By Ben Recht

Answers to four teaching-related questions from Teaching Affairs at Reykjavik University

from Luca Aceto

The Teaching Affairs Office at Reykjavik University is creating a video in which several faculty members answer some questions related to the principles that guide their pedagogical work. I am taking part in that enterprise and was asked to answer the four questions below. Here are the written versions of the answers I intended to give orally, as a note to myself and in case they are of interest to anyone. 
What would your answers to those questions be? 
Q1: What is your teaching philosophy? or What is your approach to teaching? 
My approach to teaching is eclectic. Overall, I try (and often fail, alas) to create a stimulating learning environment in cooperation with my students and teaching assistants, where anyone should feel safe to make mistakes and learn from them. 
I'll do anything to entice my students to engage actively with the material covered in my courses provided it is in reasonably good taste and conveys my enthusiasm for the subject I'm asked to teach. This includes the much-maligned lecture, since I firmly believe that telling a good story is still one of the best ways we have to inspire our students and to provide the context and intellectual history of the ideas we cover in our courses. However, I view lectures as arenas for a game of intellectual table tennis with the students, as "commedia dell'arte" performances in which we interact and learn from one another. 
Q2: Has your teaching philosophy changed? If so, how and why has it changed? 
At the beginning of my career, I focused too much on the content I thought I was expected to deliver during my courses. However, over the years, I have come to realise that less is more and that we should "distil and conquer". Today, students can access content from a huge variety of sources, but they can't typically get context and enthusiasm for the subject they are learning, which, to my mind, comes from the development of a good story as a course progresses. I also think that we should try to avoid being boring in our teaching. This is especially true when we teach the topics we love in theoretical computer science. Students find those topics exotic, hard and dry, whereas our subject has a long and extremely interesting intellectual heritage that every cultured person should know and that we could covey to our students as keen storytellers. See Scott Aaronson's book and his teaching statement, which I still find inspiring after all these years. 
Q3: How do you promote active learning in your teaching? 
I try to emphasise learning in every component of my teaching. Note that I wrote "learning" because I believe that all learning is an active process. 
Even when I lecture, I do my best to play intellectual table tennis with my students by asking them "why" and "what if" questions, and by encouraging them to think about the concepts we are covering in real time. Apart from using the Socratic method in my lecturing, I set students assignments so that they can deepen their understanding of the course material by practising the skills they have learnt. In several of my courses, I also give students a couple of open-ended group projects that are just beyond their current abilities, so that they can challenge themselves and engage in peer learning. 
The most extreme form of active learning I have used in my teaching is exemplified by an intensive three-week course I designed with Anna Ingolfsdottir and that I have taught each year in the period 2013-2023. After a brief introduction putting the material the students are going to learn and the work they are going to do in context, I give them material they should read and start experimenting with independently by the end of day one of the course. From day two of the course and over the following three weeks, I set students challenges that they tackle in groups at their own speed, acting as a facilitator in the classroom and introducing new course topics on a by-need basis. 
Every year, students rise to the challenge and do work which often goes beyond my expectations and of which they are proud. 
Q4: What have you found to be the most effective active learning strategy in your teaching context?
I don't believe that there is a "most effective active learning strategy". However, having worked at Aalborg University for a decade in a previous life, I have come to appreciate the Aalborg model for problem-based learning as an extremely beneficial strategy to foster creativity, critical thinking and the ability to learn independently in students. To my mind, Reykjavik University would stand to gain by embracing more aspects of problem-based learning in earnest. 

By Luca Aceto

The Teaching Affairs Office at Reykjavik University is creating a video in which several faculty members answer some questions related to the principles that guide their pedagogical work. I am taking part in that enterprise and was asked to answer the four questions below. Here are the written versions of the answers I intended to give orally, as a note to myself and in case they are of interest to anyone. 

What would your answers to those questions be? 

Q1: What is your teaching philosophy? or What is your approach to teaching? 

My approach to teaching is eclectic. Overall, I try (and often fail, alas) to create a stimulating learning environment in cooperation with my students and teaching assistants, where anyone should feel safe to make mistakes and learn from them. 

I'll do anything to entice my students to engage actively with the material covered in my courses provided it is in reasonably good taste and conveys my enthusiasm for the subject I'm asked to teach. This includes the much-maligned lecture, since I firmly believe that telling a good story is still one of the best ways we have to inspire our students and to provide the context and intellectual history of the ideas we cover in our courses. However, I view lectures as arenas for a game of intellectual table tennis with the students, as "commedia dell'arte" performances in which we interact and learn from one another. 

Q2: Has your teaching philosophy changed? If so, how and why has it changed? 

At the beginning of my career, I focused too much on the content I thought I was expected to deliver during my courses. However, over the years, I have come to realise that less is more and that we should "distil and conquer". Today, students can access content from a huge variety of sources, but they can't typically get context and enthusiasm for the subject they are learning, which, to my mind, comes from the development of a good story as a course progresses. I also think that we should try to avoid being boring in our teaching. This is especially true when we teach the topics we love in theoretical computer science. Students find those topics exotic, hard and dry, whereas our subject has a long and extremely interesting intellectual heritage that every cultured person should know and that we could covey to our students as keen storytellers. See Scott Aaronson's book and his teaching statement, which I still find inspiring after all these years. 

Q3: How do you promote active learning in your teaching? 

I try to emphasise learning in every component of my teaching. Note that I wrote "learning" because I believe that all learning is an active process. 

Even when I lecture, I do my best to play intellectual table tennis with my students by asking them "why" and "what if" questions, and by encouraging them to think about the concepts we are covering in real time. Apart from using the Socratic method in my lecturing, I set students assignments so that they can deepen their understanding of the course material by practising the skills they have learnt. In several of my courses, I also give students a couple of open-ended group projects that are just beyond their current abilities, so that they can challenge themselves and engage in peer learning. 

The most extreme form of active learning I have used in my teaching is exemplified by an intensive three-week course I designed with Anna Ingolfsdottir and that I have taught each year in the period 2013-2023. After a brief introduction putting the material the students are going to learn and the work they are going to do in context, I give them material they should read and start experimenting with independently by the end of day one of the course. From day two of the course and over the following three weeks, I set students challenges that they tackle in groups at their own speed, acting as a facilitator in the classroom and introducing new course topics on a by-need basis. 

Every year, students rise to the challenge and do work which often goes beyond my expectations and of which they are proud. 

Q4: What have you found to be the most effective active learning strategy in your teaching context?

I don't believe that there is a "most effective active learning strategy". However, having worked at Aalborg University for a decade in a previous life, I have come to appreciate the Aalborg model for problem-based learning as an extremely beneficial strategy to foster creativity, critical thinking and the ability to learn independently in students. To my mind, Reykjavik University would stand to gain by embracing more aspects of problem-based learning in earnest. 

By Luca Aceto

TR25-052 | Deterministic Depth-4 PIT and Normalization | Zeyu Guo, Siki Wang

from ECCC Papers

In this paper, we initiate the study of deterministic PIT for $\Sigma^{[k]}\Pi\Sigma\Pi^{[\delta]}$ circuits over fields of any characteristic, where $k$ and $\delta$ are bounded. Our main result is a deterministic polynomial-time black-box PIT algorithm for $\Sigma^{[3]}\Pi\Sigma\Pi^{[\delta]}$ circuits, under the additional condition that one of the summands at the top $\Sigma$ gate is squarefree. Our techniques are purely algebro-geometric: they do not rely on Sylvester--Gallai-type theorems, and our PIT result holds over arbitrary fields. The core of our proof is based on the normalization of algebraic varieties. Specifically, we carry out the analysis in the integral closure of a coordinate ring, which enjoys better algebraic properties than the original ring.

In this paper, we initiate the study of deterministic PIT for $\Sigma^{[k]}\Pi\Sigma\Pi^{[\delta]}$ circuits over fields of any characteristic, where $k$ and $\delta$ are bounded. Our main result is a deterministic polynomial-time black-box PIT algorithm for $\Sigma^{[3]}\Pi\Sigma\Pi^{[\delta]}$ circuits, under the additional condition that one of the summands at the top $\Sigma$ gate is squarefree. Our techniques are purely algebro-geometric: they do not rely on Sylvester--Gallai-type theorems, and our PIT result holds over arbitrary fields. The core of our proof is based on the normalization of algebraic varieties. Specifically, we carry out the analysis in the integral closure of a coordinate ring, which enjoys better algebraic properties than the original ring.

How Global Calibration Strengthens Multiaccuracy

from arXiv: Computational Complexity

Authors: Sílvia Casacuberta, Parikshit Gopalan, Varun Kanade, Omer Reingold

Multiaccuracy and multicalibration are multigroup fairness notions for prediction that have found numerous applications in learning and computational complexity. They can be achieved from a single learning primitive: weak agnostic learning. Here we investigate the power of multiaccuracy as a learning primitive, both with and without the additional assumption of calibration. We find that multiaccuracy in itself is rather weak, but that the addition of global calibration (this notion is called calibrated multiaccuracy) boosts its power substantially, enough to recover implications that were previously known only assuming the stronger notion of multicalibration. We give evidence that multiaccuracy might not be as powerful as standard weak agnostic learning, by showing that there is no way to post-process a multiaccurate predictor to get a weak learner, even assuming the best hypothesis has correlation $1/2$. Rather, we show that it yields a restricted form of weak agnostic learning, which requires some concept in the class to have correlation greater than $1/2$ with the labels. However, by also requiring the predictor to be calibrated, we recover not just weak, but strong agnostic learning. A similar picture emerges when we consider the derivation of hardcore measures from predictors satisfying multigroup fairness notions. On the one hand, while multiaccuracy only yields hardcore measures of density half the optimal, we show that (a weighted version of) calibrated multiaccuracy achieves optimal density. Our results yield new insights into the complementary roles played by multiaccuracy and calibration in each setting. They shed light on why multiaccuracy and global calibration, although not particularly powerful by themselves, together yield considerably stronger notions.

Authors: Sílvia Casacuberta, Parikshit Gopalan, Varun Kanade, Omer Reingold

Multiaccuracy and multicalibration are multigroup fairness notions for prediction that have found numerous applications in learning and computational complexity. They can be achieved from a single learning primitive: weak agnostic learning. Here we investigate the power of multiaccuracy as a learning primitive, both with and without the additional assumption of calibration. We find that multiaccuracy in itself is rather weak, but that the addition of global calibration (this notion is called calibrated multiaccuracy) boosts its power substantially, enough to recover implications that were previously known only assuming the stronger notion of multicalibration. We give evidence that multiaccuracy might not be as powerful as standard weak agnostic learning, by showing that there is no way to post-process a multiaccurate predictor to get a weak learner, even assuming the best hypothesis has correlation $1/2$. Rather, we show that it yields a restricted form of weak agnostic learning, which requires some concept in the class to have correlation greater than $1/2$ with the labels. However, by also requiring the predictor to be calibrated, we recover not just weak, but strong agnostic learning. A similar picture emerges when we consider the derivation of hardcore measures from predictors satisfying multigroup fairness notions. On the one hand, while multiaccuracy only yields hardcore measures of density half the optimal, we show that (a weighted version of) calibrated multiaccuracy achieves optimal density. Our results yield new insights into the complementary roles played by multiaccuracy and calibration in each setting. They shed light on why multiaccuracy and global calibration, although not particularly powerful by themselves, together yield considerably stronger notions.

Deterministic Depth-4 PIT and Normalization

from arXiv: Computational Complexity

Authors: Zeyu Guo, Siki Wang

In this paper, we initiate the study of deterministic PIT for $\Sigma^{[k]}\Pi\Sigma\Pi^{[\delta]}$ circuits over fields of any characteristic, where $k$ and $\delta$ are bounded. Our main result is a deterministic polynomial-time black-box PIT algorithm for $\Sigma^{[3]}\Pi\Sigma\Pi^{[\delta]}$ circuits, under the additional condition that one of the summands at the top $\Sigma$ gate is squarefree. Our techniques are purely algebro-geometric: they do not rely on Sylvester--Gallai-type theorems, and our PIT result holds over arbitrary fields. The core of our proof is based on the normalization of algebraic varieties. Specifically, we carry out the analysis in the integral closure of a coordinate ring, which enjoys better algebraic properties than the original ring.

Authors: Zeyu Guo, Siki Wang

In this paper, we initiate the study of deterministic PIT for $\Sigma^{[k]}\Pi\Sigma\Pi^{[\delta]}$ circuits over fields of any characteristic, where $k$ and $\delta$ are bounded. Our main result is a deterministic polynomial-time black-box PIT algorithm for $\Sigma^{[3]}\Pi\Sigma\Pi^{[\delta]}$ circuits, under the additional condition that one of the summands at the top $\Sigma$ gate is squarefree. Our techniques are purely algebro-geometric: they do not rely on Sylvester--Gallai-type theorems, and our PIT result holds over arbitrary fields. The core of our proof is based on the normalization of algebraic varieties. Specifically, we carry out the analysis in the integral closure of a coordinate ring, which enjoys better algebraic properties than the original ring.

Parallel Kac's Walk Generates PRU

from arXiv: Computational Complexity

Authors: Chuhan Lu, Minglong Qin, Fang Song, Penghui Yao, Mingnan Zhao

Ma and Huang recently proved that the PFC construction, introduced by Metger, Poremba, Sinha and Yuen [MPSY24], gives an adaptive-secure pseudorandom unitary family PRU. Their proof developed a new path recording technique [MH24]. In this work, we show that a linear number of sequential repetitions of the parallel Kac's Walk, introduced by Lu, Qin, Song, Yao and Zhao [LQSY+24], also forms an adaptive-secure PRU, confirming a conjecture therein. Moreover, it additionally satisfies strong security against adversaries making inverse queries. This gives an alternative PRU construction, and provides another instance demonstrating the power of the path recording technique. We also discuss some further simplifications and implications.

Authors: Chuhan Lu, Minglong Qin, Fang Song, Penghui Yao, Mingnan Zhao

Ma and Huang recently proved that the PFC construction, introduced by Metger, Poremba, Sinha and Yuen [MPSY24], gives an adaptive-secure pseudorandom unitary family PRU. Their proof developed a new path recording technique [MH24]. In this work, we show that a linear number of sequential repetitions of the parallel Kac's Walk, introduced by Lu, Qin, Song, Yao and Zhao [LQSY+24], also forms an adaptive-secure PRU, confirming a conjecture therein. Moreover, it additionally satisfies strong security against adversaries making inverse queries. This gives an alternative PRU construction, and provides another instance demonstrating the power of the path recording technique. We also discuss some further simplifications and implications.

(Sub)Exponential Quantum Speedup for Optimization

from arXiv: Computational Complexity

Authors: Jiaqi Leng, Kewen Wu, Xiaodi Wu, Yufan Zheng

We demonstrate provable (sub)exponential quantum speedups in both discrete and continuous optimization, achieved through simple and natural quantum optimization algorithms, namely the quantum adiabatic algorithm for discrete optimization and quantum Hamiltonian descent for continuous optimization. Our result builds on the Gily\'en--Hastings--Vazirani (sub)exponential oracle separation for adiabatic quantum computing. With a sequence of perturbative reductions, we compile their construction into two standalone objective functions, whose oracles can be directly leveraged by the plain adiabatic evolution and Schr\"odinger operator evolution for discrete and continuous optimization, respectively.

Authors: Jiaqi Leng, Kewen Wu, Xiaodi Wu, Yufan Zheng

We demonstrate provable (sub)exponential quantum speedups in both discrete and continuous optimization, achieved through simple and natural quantum optimization algorithms, namely the quantum adiabatic algorithm for discrete optimization and quantum Hamiltonian descent for continuous optimization. Our result builds on the Gily\'en--Hastings--Vazirani (sub)exponential oracle separation for adiabatic quantum computing. With a sequence of perturbative reductions, we compile their construction into two standalone objective functions, whose oracles can be directly leveraged by the plain adiabatic evolution and Schr\"odinger operator evolution for discrete and continuous optimization, respectively.

A Note on the Complexity of Defensive Domination

from arXiv: Computational Complexity

Authors: Steven Chaplick, Grzegorz Gutowski, Tomasz Krawczyk

In a graph G, a k-attack A is any set of at most k vertices and l-defense D is a set of at most l vertices. We say that defense D counters attack A if each a in A can be matched to a distinct defender d in D with a equal to d or a adjacent to d in G. In the defensive domination problem, we are interested in deciding, for a graph G and positive integers k and l given on input, if there exists an l-defense that counters every possible k-attack on G. Defensive domination is a natural resource allocation problem and can be used to model network robustness and security, disaster response strategies, and redundancy designs. The defensive domination problem is naturally in the complexity class $\Sigma^P_2$. The problem was known to be NP-hard in general, and polynomial-time algorithms were found for some restricted graph classes. In this note we prove that the defensive domination problem is $\Sigma^P_2$-complete. We also introduce a natural variant of the defensive domination problem in which the defense is allowed to be a multiset of vertices. This variant is also $\Sigma^P_2$-complete, but we show that it admits a polynomial-time algorithm in the class of interval graphs. A similar result was known for the original setting in the class of proper interval graphs.

Authors: Steven Chaplick, Grzegorz Gutowski, Tomasz Krawczyk

In a graph G, a k-attack A is any set of at most k vertices and l-defense D is a set of at most l vertices. We say that defense D counters attack A if each a in A can be matched to a distinct defender d in D with a equal to d or a adjacent to d in G. In the defensive domination problem, we are interested in deciding, for a graph G and positive integers k and l given on input, if there exists an l-defense that counters every possible k-attack on G. Defensive domination is a natural resource allocation problem and can be used to model network robustness and security, disaster response strategies, and redundancy designs. The defensive domination problem is naturally in the complexity class $\Sigma^P_2$. The problem was known to be NP-hard in general, and polynomial-time algorithms were found for some restricted graph classes. In this note we prove that the defensive domination problem is $\Sigma^P_2$-complete. We also introduce a natural variant of the defensive domination problem in which the defense is allowed to be a multiset of vertices. This variant is also $\Sigma^P_2$-complete, but we show that it admits a polynomial-time algorithm in the class of interval graphs. A similar result was known for the original setting in the class of proper interval graphs.

Maker-Maker games of rank 4 are PSPACE-complete

from arXiv: Computational Complexity

Authors: Florian Galliot, Jonas Sénizergues

The Maker-Maker convention of positional games is played on a hypergraph whose edges are interpreted as winning sets. Two players take turns picking a previously unpicked vertex, aiming at being first to pick all the vertices of some edge. Optimal play can only lead to a first player win or a draw, and deciding between the two is known to be PSPACE-complete even for 6-uniform hypergraphs. We establish PSPACE-completeness for hypergraphs of rank 4. As an intermediary, we use the recently introduced achievement positional games, a more general convention in which each player has their own winning sets (blue and red). We show that deciding whether the blue player has a winning strategy as the first player is PSPACE-complete even with blue edges of size 2 or 3 and pairwise disjoint red edges of size 2. The result for hypergraphs of rank 4 in the Maker-Maker convention follows as a simple corollary.

Authors: Florian Galliot, Jonas Sénizergues

The Maker-Maker convention of positional games is played on a hypergraph whose edges are interpreted as winning sets. Two players take turns picking a previously unpicked vertex, aiming at being first to pick all the vertices of some edge. Optimal play can only lead to a first player win or a draw, and deciding between the two is known to be PSPACE-complete even for 6-uniform hypergraphs. We establish PSPACE-completeness for hypergraphs of rank 4. As an intermediary, we use the recently introduced achievement positional games, a more general convention in which each player has their own winning sets (blue and red). We show that deciding whether the blue player has a winning strategy as the first player is PSPACE-complete even with blue edges of size 2 or 3 and pairwise disjoint red edges of size 2. The result for hypergraphs of rank 4 in the Maker-Maker convention follows as a simple corollary.

Holant* Dichotomy on Domain Size 3: A Geometric Perspective

from arXiv: Computational Complexity

Authors: Jin-Yi Cai, Jin Soo Ihm

Holant problems are a general framework to study the computational complexity of counting problems. It is a more expressive framework than counting constraint satisfaction problems (CSP) which are in turn more expressive than counting graph homomorphisms (GH). In this paper, we prove the first complexity dichotomy of $\mathrm{Holant}_3(\mathcal{F})$ where $\mathcal{F}$ is an arbitrary set of symmetric, real valued constraint functions on domain size $3$. We give an explicit tractability criterion and prove that, if $\mathcal{F}$ satisfies this criterion then $\mathrm{Holant}_3(\mathcal{F})$ is polynomial time computable, and otherwise it is \#P-hard, with no intermediate cases. We show that the geometry of the tensor decomposition of the constraint functions plays a central role in the formulation as well as the structural internal logic of the dichotomy.

Authors: Jin-Yi Cai, Jin Soo Ihm

Holant problems are a general framework to study the computational complexity of counting problems. It is a more expressive framework than counting constraint satisfaction problems (CSP) which are in turn more expressive than counting graph homomorphisms (GH). In this paper, we prove the first complexity dichotomy of $\mathrm{Holant}_3(\mathcal{F})$ where $\mathcal{F}$ is an arbitrary set of symmetric, real valued constraint functions on domain size $3$. We give an explicit tractability criterion and prove that, if $\mathcal{F}$ satisfies this criterion then $\mathrm{Holant}_3(\mathcal{F})$ is polynomial time computable, and otherwise it is \#P-hard, with no intermediate cases. We show that the geometry of the tensor decomposition of the constraint functions plays a central role in the formulation as well as the structural internal logic of the dichotomy.

The Mid-sphere Cousin of the Medial Axis Transform

from arXiv: Computational Geometry

Authors: Herbert Edelsbrunner, Elizabeth Stephenson, Martin Hafskjold Thoresen

The medial axis of a smoothly embedded surface in $\mathbb{R}^3$ consists of all points for which the Euclidean distance function on the surface has at least two minima. We generalize this notion to the mid-sphere axis, which consists of all points for which the Euclidean distance function has two interchanging saddles that swap their partners in the pairing by persistent homology. It offers a discrete-algebraic multi-scale approach to computing ridge-like structures on the surface. As a proof of concept, an algorithm that computes stair-case approximations of the mid-sphere axis is provided.

Authors: Herbert Edelsbrunner, Elizabeth Stephenson, Martin Hafskjold Thoresen

The medial axis of a smoothly embedded surface in $\mathbb{R}^3$ consists of all points for which the Euclidean distance function on the surface has at least two minima. We generalize this notion to the mid-sphere axis, which consists of all points for which the Euclidean distance function has two interchanging saddles that swap their partners in the pairing by persistent homology. It offers a discrete-algebraic multi-scale approach to computing ridge-like structures on the surface. As a proof of concept, an algorithm that computes stair-case approximations of the mid-sphere axis is provided.

Leibniz rule for wedge product in discrete exterior calculus on general polygonal meshes

from arXiv: Computational Geometry

Authors: Lenka Ptackova

Discrete exterior calculus offers a coordinate-free discretization of exterior calculus especially suited for computations on curved spaces. In this work, we present a wedge product on 2-dimensional pseudomanifolds, whose faces are any polygons. We prove that this polygonal wedge product is compatible with the discrete exterior derivative in the sense that it satisfies the Leibniz product rule. We thus extend previously studied discretizations of wedge products from simplicial or quadrilateral meshes to general polygonal surface meshes. We also prove that our discrete wedge product corresponds to a cup product of cochains on 2-pseudomanifolds.

Authors: Lenka Ptackova

Discrete exterior calculus offers a coordinate-free discretization of exterior calculus especially suited for computations on curved spaces. In this work, we present a wedge product on 2-dimensional pseudomanifolds, whose faces are any polygons. We prove that this polygonal wedge product is compatible with the discrete exterior derivative in the sense that it satisfies the Leibniz product rule. We thus extend previously studied discretizations of wedge products from simplicial or quadrilateral meshes to general polygonal surface meshes. We also prove that our discrete wedge product corresponds to a cup product of cochains on 2-pseudomanifolds.

Explicit Lossless Vertex Expanders

from arXiv: Data Structures and Algorithms

Authors: Jun-Ting Hsieh, Alexander Lubotzky, Sidhanth Mohanty, Assaf Reiner, Rachel Yun Zhang

We give the first construction of explicit constant-degree lossless vertex expanders. Specifically, for any $\varepsilon > 0$ and sufficiently large $d$, we give an explicit construction of an infinite family of $d$-regular graphs where every small set $S$ of vertices has $(1-\varepsilon)d|S|$ neighbors (which implies $(1-2\varepsilon)d|S|$ unique-neighbors). Our results also extend naturally to construct biregular bipartite graphs of any constant imbalance, where small sets on each side have strong expansion guarantees. The graphs we construct admit a free group action, and hence realize new families of quantum LDPC codes of Lin and M. Hsieh with a linear time decoding algorithm. Our construction is based on taking an appropriate product of a constant-sized lossless expander with a base graph constructed from Ramanujan Cayley cubical complexes.

Authors: Jun-Ting Hsieh, Alexander Lubotzky, Sidhanth Mohanty, Assaf Reiner, Rachel Yun Zhang

We give the first construction of explicit constant-degree lossless vertex expanders. Specifically, for any $\varepsilon > 0$ and sufficiently large $d$, we give an explicit construction of an infinite family of $d$-regular graphs where every small set $S$ of vertices has $(1-\varepsilon)d|S|$ neighbors (which implies $(1-2\varepsilon)d|S|$ unique-neighbors). Our results also extend naturally to construct biregular bipartite graphs of any constant imbalance, where small sets on each side have strong expansion guarantees. The graphs we construct admit a free group action, and hence realize new families of quantum LDPC codes of Lin and M. Hsieh with a linear time decoding algorithm. Our construction is based on taking an appropriate product of a constant-sized lossless expander with a base graph constructed from Ramanujan Cayley cubical complexes.

Approximate all-pairs Hamming distances and 0-1 matrix multiplication

from arXiv: Data Structures and Algorithms

Authors: Miroslaw Kowaluk, Andrzej Lingas, Mia Persson

Arslan showed that computing all-pairs Hamming distances is easily reducible to arithmetic 0-1 matrix multiplication (IPL 2018). We provide a reverse, linear-time reduction of arithmetic 0-1 matrix multiplication to computing all-pairs distances in a Hamming space. On the other hand, we present a fast randomized algorithm for approximate all-pairs distances in a Hamming space. By combining it with our reduction, we obtain also a fast randomized algorithm for approximate 0-1 matrix multiplication. Next, we present an output-sensitive randomized algorithm for a minimum spanning tree of a set of points in a generalized Hamming space, the lower is the cost of the minimum spanning tree the faster is our algorithm. Finally, we provide $(2+\epsilon)$- approximation algorithms for the $\ell$-center clustering and minimum-diameter $\ell$-clustering problems in a Hamming space $\{0,1\}^d$ that are substantially faster than the known $2$-approximation ones when both $\ell$ and $d$ are super-logarithmic.

Authors: Miroslaw Kowaluk, Andrzej Lingas, Mia Persson

Arslan showed that computing all-pairs Hamming distances is easily reducible to arithmetic 0-1 matrix multiplication (IPL 2018). We provide a reverse, linear-time reduction of arithmetic 0-1 matrix multiplication to computing all-pairs distances in a Hamming space. On the other hand, we present a fast randomized algorithm for approximate all-pairs distances in a Hamming space. By combining it with our reduction, we obtain also a fast randomized algorithm for approximate 0-1 matrix multiplication. Next, we present an output-sensitive randomized algorithm for a minimum spanning tree of a set of points in a generalized Hamming space, the lower is the cost of the minimum spanning tree the faster is our algorithm. Finally, we provide $(2+\epsilon)$- approximation algorithms for the $\ell$-center clustering and minimum-diameter $\ell$-clustering problems in a Hamming space $\{0,1\}^d$ that are substantially faster than the known $2$-approximation ones when both $\ell$ and $d$ are super-logarithmic.

On Learning Parallel Pancakes with Mostly Uniform Weights

from arXiv: Data Structures and Algorithms

Authors: Ilias Diakonikolas, Daniel M. Kane, Sushrut Karmalkar, Jasper C. H. Lee, Thanasis Pittas

We study the complexity of learning $k$-mixtures of Gaussians ($k$-GMMs) on $\mathbb{R}^d$. This task is known to have complexity $d^{\Omega(k)}$ in full generality. To circumvent this exponential lower bound on the number of components, research has focused on learning families of GMMs satisfying additional structural properties. A natural assumption posits that the component weights are not exponentially small and that the components have the same unknown covariance. Recent work gave a $d^{O(\log(1/w_{\min}))}$-time algorithm for this class of GMMs, where $w_{\min}$ is the minimum weight. Our first main result is a Statistical Query (SQ) lower bound showing that this quasi-polynomial upper bound is essentially best possible, even for the special case of uniform weights. Specifically, we show that it is SQ-hard to distinguish between such a mixture and the standard Gaussian. We further explore how the distribution of weights affects the complexity of this task. Our second main result is a quasi-polynomial upper bound for the aforementioned testing task when most of the weights are uniform while a small fraction of the weights are potentially arbitrary.

Authors: Ilias Diakonikolas, Daniel M. Kane, Sushrut Karmalkar, Jasper C. H. Lee, Thanasis Pittas

We study the complexity of learning $k$-mixtures of Gaussians ($k$-GMMs) on $\mathbb{R}^d$. This task is known to have complexity $d^{\Omega(k)}$ in full generality. To circumvent this exponential lower bound on the number of components, research has focused on learning families of GMMs satisfying additional structural properties. A natural assumption posits that the component weights are not exponentially small and that the components have the same unknown covariance. Recent work gave a $d^{O(\log(1/w_{\min}))}$-time algorithm for this class of GMMs, where $w_{\min}$ is the minimum weight. Our first main result is a Statistical Query (SQ) lower bound showing that this quasi-polynomial upper bound is essentially best possible, even for the special case of uniform weights. Specifically, we show that it is SQ-hard to distinguish between such a mixture and the standard Gaussian. We further explore how the distribution of weights affects the complexity of this task. Our second main result is a quasi-polynomial upper bound for the aforementioned testing task when most of the weights are uniform while a small fraction of the weights are potentially arbitrary.

Faster Algorithms for Agnostically Learning Disjunctions and their Implications

from arXiv: Data Structures and Algorithms

Authors: Ilias Diakonikolas, Daniel M. Kane, Lisheng Ren

We study the algorithmic task of learning Boolean disjunctions in the distribution-free agnostic PAC model. The best known agnostic learner for the class of disjunctions over $\{0, 1\}^n$ is the $L_1$-polynomial regression algorithm, achieving complexity $2^{\tilde{O}(n^{1/2})}$. This complexity bound is known to be nearly best possible within the class of Correlational Statistical Query (CSQ) algorithms. In this work, we develop an agnostic learner for this concept class with complexity $2^{\tilde{O}(n^{1/3})}$. Our algorithm can be implemented in the Statistical Query (SQ) model, providing the first separation between the SQ and CSQ models in distribution-free agnostic learning.

Authors: Ilias Diakonikolas, Daniel M. Kane, Lisheng Ren

We study the algorithmic task of learning Boolean disjunctions in the distribution-free agnostic PAC model. The best known agnostic learner for the class of disjunctions over $\{0, 1\}^n$ is the $L_1$-polynomial regression algorithm, achieving complexity $2^{\tilde{O}(n^{1/2})}$. This complexity bound is known to be nearly best possible within the class of Correlational Statistical Query (CSQ) algorithms. In this work, we develop an agnostic learner for this concept class with complexity $2^{\tilde{O}(n^{1/3})}$. Our algorithm can be implemented in the Statistical Query (SQ) model, providing the first separation between the SQ and CSQ models in distribution-free agnostic learning.

Distribution Testing Meets Sum Estimation

from arXiv: Data Structures and Algorithms

Authors: Pinki Pradhan, Sampriti Roy

We study the problem of estimating the sum of $n$ elements, each with weight $w(i)$, in a structured universe. Our goal is to estimate $W = \sum_{i=1}^n w(i)$ within a $(1 \pm \epsilon)$ factor using a sublinear number of samples, assuming weights are non-increasing, i.e., $w(1) \geq w(2) \geq \dots \geq w(n)$. The sum estimation problem is well-studied under different access models to the universe $U$. However, to the best of our knowledge, nothing is known about the sum estimation problem using non-adaptive conditional sampling. In this work, we explore the sum estimation problem using non-adaptive conditional weighted and non-adaptive conditional uniform samples, assuming that the underlying distribution ($D(i)=w(i)/W$) is monotone. We also extend our approach to to the case where the underlying distribution of $U$ is unimodal. Additionally, we consider support size estimation when $w(i) = 0$ or $w(i) \geq W/n$, using hybrid sampling (both weighted and uniform) to access $U$. We propose an algorithm to estimate $W$ under the non-increasing weight assumption, using $O(\frac{1}{\epsilon^3} \log{n} + \frac{1}{\epsilon^6})$ non-adaptive weighted conditional samples and $O(\frac{1}{\epsilon^3} \log{n})$ uniform conditional samples. Our algorithm matches the $\Omega(\log{n})$ lower bound by \cite{ACK15}. For unimodal distributions, the sample complexity remains similar, with an additional $O(\log{n})$ evaluation queries to locate the minimum weighted point in the domain. For estimating the support size $k$ of $U$, where weights are either $0$ or at least $W/n$, our algorithm uses $O\big( \frac{\log^3(n/\epsilon)}{\epsilon^8} \cdot \log^4 \frac{\log(n/\epsilon)}{\epsilon} \big)$ uniform samples and $O\big( \frac{\log(n/\epsilon)}{\epsilon^2} \cdot \log \frac{\log(n/\epsilon)}{\epsilon} \big)$ weighted samples to output $\hat{k}$ satisfying $k - 2\epsilon n \leq \hat{k} \leq k + \epsilon n$.

Authors: Pinki Pradhan, Sampriti Roy

We study the problem of estimating the sum of $n$ elements, each with weight $w(i)$, in a structured universe. Our goal is to estimate $W = \sum_{i=1}^n w(i)$ within a $(1 \pm \epsilon)$ factor using a sublinear number of samples, assuming weights are non-increasing, i.e., $w(1) \geq w(2) \geq \dots \geq w(n)$. The sum estimation problem is well-studied under different access models to the universe $U$. However, to the best of our knowledge, nothing is known about the sum estimation problem using non-adaptive conditional sampling. In this work, we explore the sum estimation problem using non-adaptive conditional weighted and non-adaptive conditional uniform samples, assuming that the underlying distribution ($D(i)=w(i)/W$) is monotone. We also extend our approach to to the case where the underlying distribution of $U$ is unimodal. Additionally, we consider support size estimation when $w(i) = 0$ or $w(i) \geq W/n$, using hybrid sampling (both weighted and uniform) to access $U$. We propose an algorithm to estimate $W$ under the non-increasing weight assumption, using $O(\frac{1}{\epsilon^3} \log{n} + \frac{1}{\epsilon^6})$ non-adaptive weighted conditional samples and $O(\frac{1}{\epsilon^3} \log{n})$ uniform conditional samples. Our algorithm matches the $\Omega(\log{n})$ lower bound by \cite{ACK15}. For unimodal distributions, the sample complexity remains similar, with an additional $O(\log{n})$ evaluation queries to locate the minimum weighted point in the domain. For estimating the support size $k$ of $U$, where weights are either $0$ or at least $W/n$, our algorithm uses $O\big( \frac{\log^3(n/\epsilon)}{\epsilon^8} \cdot \log^4 \frac{\log(n/\epsilon)}{\epsilon} \big)$ uniform samples and $O\big( \frac{\log(n/\epsilon)}{\epsilon^2} \cdot \log \frac{\log(n/\epsilon)}{\epsilon} \big)$ weighted samples to output $\hat{k}$ satisfying $k - 2\epsilon n \leq \hat{k} \leq k + \epsilon n$.

Deterministic $k$-Median Clustering in Near-Optimal Time

from arXiv: Data Structures and Algorithms

Authors: Martín Costa, Ermiya Farokhnejad

The metric $k$-median problem is a textbook clustering problem. As input, we are given a metric space $V$ of size $n$ and an integer $k$, and our task is to find a subset $S \subseteq V$ of at most $k$ `centers' that minimizes the total distance from each point in $V$ to its nearest center in $S$. Mettu and Plaxton [UAI'02] gave a randomized algorithm for $k$-median that computes a $O(1)$-approximation in $\tilde O(nk)$ time. They also showed that any algorithm for this problem with a bounded approximation ratio must have a running time of $\Omega(nk)$. Thus, the running time of their algorithm is optimal up to polylogarithmic factors. For deterministic $k$-median, Guha et al.~[FOCS'00] gave an algorithm that computes a $\text{poly}(\log (n/k))$-approximation in $\tilde O(nk)$ time, where the degree of the polynomial in the approximation is unspecified. To the best of our knowledge, this remains the state-of-the-art approximation of any deterministic $k$-median algorithm with this running time. This leads us to the following natural question: What is the best approximation of a deterministic $k$-median algorithm with near-optimal running time? We make progress in answering this question by giving a deterministic algorithm that computes a $O(\log(n/k))$-approximation in $\tilde O(nk)$ time. We also provide a lower bound showing that any deterministic algorithm with this running time must have an approximation ratio of $\Omega(\log n/(\log k + \log \log n))$, establishing a gap between the randomized and deterministic settings for $k$-median.

Authors: Martín Costa, Ermiya Farokhnejad

The metric $k$-median problem is a textbook clustering problem. As input, we are given a metric space $V$ of size $n$ and an integer $k$, and our task is to find a subset $S \subseteq V$ of at most $k$ `centers' that minimizes the total distance from each point in $V$ to its nearest center in $S$. Mettu and Plaxton [UAI'02] gave a randomized algorithm for $k$-median that computes a $O(1)$-approximation in $\tilde O(nk)$ time. They also showed that any algorithm for this problem with a bounded approximation ratio must have a running time of $\Omega(nk)$. Thus, the running time of their algorithm is optimal up to polylogarithmic factors. For deterministic $k$-median, Guha et al.~[FOCS'00] gave an algorithm that computes a $\text{poly}(\log (n/k))$-approximation in $\tilde O(nk)$ time, where the degree of the polynomial in the approximation is unspecified. To the best of our knowledge, this remains the state-of-the-art approximation of any deterministic $k$-median algorithm with this running time. This leads us to the following natural question: What is the best approximation of a deterministic $k$-median algorithm with near-optimal running time? We make progress in answering this question by giving a deterministic algorithm that computes a $O(\log(n/k))$-approximation in $\tilde O(nk)$ time. We also provide a lower bound showing that any deterministic algorithm with this running time must have an approximation ratio of $\Omega(\log n/(\log k + \log \log n))$, establishing a gap between the randomized and deterministic settings for $k$-median.