Theory of Computing Report

Authors: Joshua Brakensiek, Venkatesan Guruswami, Bart M. P. Jansen, Victor Lagerkvist, Magnus Wahlström

In the field of constraint satisfaction problems (CSP), a clause is called redundant if its satisfaction is implied by satisfying all other clauses. An instance of CSP$(P)$ is called non-redundant if it does not contain any redundant clause. The non-redundancy (NRD) of a predicate $P$ is the maximum number of clauses in a non-redundant instance of CSP$(P)$, as a function of the number of variables $n$. Recent progress has shown that non-redundancy is crucially linked to many other important questions in computer science and mathematics including sparsification, kernelization, query complexity, universal algebra, and extremal combinatorics. Given that non-redundancy is a nexus for many of these important problems, the central goal of this paper is to more deeply understand non-redundancy. Our first main result shows that for every rational number $r \ge 1$, there exists a finite CSP predicate $P$ such that the non-redundancy of $P$ is $\Theta(n^r)$. Our second main result explores the concept of conditional non-redundancy first coined by Brakensiek and Guruswami [STOC 2025]. We completely classify the conditional non-redundancy of all binary predicates (i.e., constraints on two variables) by connecting these non-redundancy problems to the structure of high-girth graphs in extremal combinatorics. Inspired by these concrete results, we build off the work of Carbonnel [CP 2022] to develop an algebraic theory of conditional non-redundancy. As an application of this algebraic theory, we revisit the notion of Mal'tsev embeddings, which is the most general technique known to date for establishing that a predicate has linear non-redundancy. For example, we provide the first example of predicate with a Mal'tsev embedding that cannot be attributed to the structure of an Abelian group, but rather to the structure of the quantum Pauli group.

Authors: Joshua Brakensiek, Venkatesan Guruswami, Bart M. P. Jansen, Victor Lagerkvist, Magnus Wahlström

In the field of constraint satisfaction problems (CSP), a clause is called redundant if its satisfaction is implied by satisfying all other clauses. An instance of CSP$(P)$ is called non-redundant if it does not contain any redundant clause. The non-redundancy (NRD) of a predicate $P$ is the maximum number of clauses in a non-redundant instance of CSP$(P)$, as a function of the number of variables $n$. Recent progress has shown that non-redundancy is crucially linked to many other important questions in computer science and mathematics including sparsification, kernelization, query complexity, universal algebra, and extremal combinatorics. Given that non-redundancy is a nexus for many of these important problems, the central goal of this paper is to more deeply understand non-redundancy. Our first main result shows that for every rational number $r \ge 1$, there exists a finite CSP predicate $P$ such that the non-redundancy of $P$ is $\Theta(n^r)$. Our second main result explores the concept of conditional non-redundancy first coined by Brakensiek and Guruswami [STOC 2025]. We completely classify the conditional non-redundancy of all binary predicates (i.e., constraints on two variables) by connecting these non-redundancy problems to the structure of high-girth graphs in extremal combinatorics. Inspired by these concrete results, we build off the work of Carbonnel [CP 2022] to develop an algebraic theory of conditional non-redundancy. As an application of this algebraic theory, we revisit the notion of Mal'tsev embeddings, which is the most general technique known to date for establishing that a predicate has linear non-redundancy. For example, we provide the first example of predicate with a Mal'tsev embedding that cannot be attributed to the structure of an Abelian group, but rather to the structure of the quantum Pauli group.

Friday, July 11 2025, 00:00

Turing complete Navier-Stokes steady states via cosymplectic geometry

Authors: Søren Dyhr, Ángel González-Prieto, Eva Miranda, Daniel Peralta-Salas

In this article, we construct stationary solutions to the Navier-Stokes equations on certain Riemannian $3$-manifolds that exhibit Turing completeness, in the sense that they are capable of performing universal computation. This universality arises on manifolds admitting nonvanishing harmonic 1-forms, thus showing that computational universality is not obstructed by viscosity, provided the underlying geometry satisfies a mild cohomological condition. The proof makes use of a correspondence between nonvanishing harmonic $1$-forms and cosymplectic geometry, which extends the classical correspondence between Beltrami fields and Reeb flows on contact manifolds.

Authors: Søren Dyhr, Ángel González-Prieto, Eva Miranda, Daniel Peralta-Salas

In this article, we construct stationary solutions to the Navier-Stokes equations on certain Riemannian $3$-manifolds that exhibit Turing completeness, in the sense that they are capable of performing universal computation. This universality arises on manifolds admitting nonvanishing harmonic 1-forms, thus showing that computational universality is not obstructed by viscosity, provided the underlying geometry satisfies a mild cohomological condition. The proof makes use of a correspondence between nonvanishing harmonic $1$-forms and cosymplectic geometry, which extends the classical correspondence between Beltrami fields and Reeb flows on contact manifolds.

Friday, July 11 2025, 00:00

Testing Isomorphism of Boolean Functions over Finite Abelian Groups

Authors: Swarnalipa Datta, Arijit Ghosh, Chandrima Kayal, Manaswi Paraashar, Manmatha Roy

Let $f$ and $g$ be Boolean functions over a finite Abelian group $\mathcal{G}$, where $g$ is fully known, and we have {\em query access} to $f$, that is, given any $x \in \mathcal{G}$ we can get the value $f(x)$. We study the tolerant isomorphism testing problem: given $\epsilon \geq 0$ and $\tau > 0$, we seek to determine, with minimal queries, whether there exists an automorphism $\sigma$ of $\mathcal{G}$ such that the fractional Hamming distance between $f \circ \sigma$ and $g$ is at most $\epsilon$, or whether for all automorphisms $\sigma$, the distance is at least $\epsilon + \tau$. We design an efficient tolerant testing algorithm for this problem, with query complexity $\mathrm{poly}\left( s, 1/\tau \right)$, where $s$ bounds the spectral norm of $g$. Additionally, we present an improved algorithm when $g$ is Fourier sparse. Our approach uses key concepts from Abelian group theory and Fourier analysis, including the annihilator of a subgroup, Pontryagin duality, and a pseudo inner-product for finite Abelian groups. We believe these techniques will find further applications in property testing.

Authors: Swarnalipa Datta, Arijit Ghosh, Chandrima Kayal, Manaswi Paraashar, Manmatha Roy

Let $f$ and $g$ be Boolean functions over a finite Abelian group $\mathcal{G}$, where $g$ is fully known, and we have {\em query access} to $f$, that is, given any $x \in \mathcal{G}$ we can get the value $f(x)$. We study the tolerant isomorphism testing problem: given $\epsilon \geq 0$ and $\tau > 0$, we seek to determine, with minimal queries, whether there exists an automorphism $\sigma$ of $\mathcal{G}$ such that the fractional Hamming distance between $f \circ \sigma$ and $g$ is at most $\epsilon$, or whether for all automorphisms $\sigma$, the distance is at least $\epsilon + \tau$. We design an efficient tolerant testing algorithm for this problem, with query complexity $\mathrm{poly}\left( s, 1/\tau \right)$, where $s$ bounds the spectral norm of $g$. Additionally, we present an improved algorithm when $g$ is Fourier sparse. Our approach uses key concepts from Abelian group theory and Fourier analysis, including the annihilator of a subgroup, Pontryagin duality, and a pseudo inner-product for finite Abelian groups. We believe these techniques will find further applications in property testing.

Friday, July 11 2025, 00:00

Nonogram: Complexity of Inference and Phase Transition Behavior

Authors: Aaron Foote, Danny Krizanc

Nonogram is a popular combinatorial puzzle (similar in nature to Sudoku or Minesweeper) in which a puzzle solver must determine if there exists a setting of the puzzle parameters that satisfy a given set of constraints. It has long been known that the problem of deciding if a solution exists is a computationally difficult problem. Despite this fact, humans still seem to enjoy playing it. This work aims to reconcile these seemingly contradictory facts by (1) analyzing the complexity of the inference problem for Nonogram (the problem of determining if there exists a puzzle parameter that can be inferred from the constraints without guessing) and (2) experimentally establishing the existence of a phase transition behavior for this inference problem. Our results show that the difficulty of the inference problem is largely determined by the density of filled cells (positive parameters) in a given puzzle. Along the way we implement an efficient encoding of a Nonogram board as a Boolean formula in Conjunctive Normal Form (CNF) through the use of regular expressions in order to make our experiments feasible.

Authors: Aaron Foote, Danny Krizanc

Nonogram is a popular combinatorial puzzle (similar in nature to Sudoku or Minesweeper) in which a puzzle solver must determine if there exists a setting of the puzzle parameters that satisfy a given set of constraints. It has long been known that the problem of deciding if a solution exists is a computationally difficult problem. Despite this fact, humans still seem to enjoy playing it. This work aims to reconcile these seemingly contradictory facts by (1) analyzing the complexity of the inference problem for Nonogram (the problem of determining if there exists a puzzle parameter that can be inferred from the constraints without guessing) and (2) experimentally establishing the existence of a phase transition behavior for this inference problem. Our results show that the difficulty of the inference problem is largely determined by the density of filled cells (positive parameters) in a given puzzle. Along the way we implement an efficient encoding of a Nonogram board as a Boolean formula in Conjunctive Normal Form (CNF) through the use of regular expressions in order to make our experiments feasible.

Friday, July 11 2025, 00:00

Approximation Depth of Convex Polytopes

Authors: Egor Bakaev, Florestan Brunck, Amir Yehudayoff

We study approximations of polytopes in the standard model for computing polytopes using Minkowski sums and (convex hulls of) unions. Specifically, we study the ability to approximate a target polytope by polytopes of a given depth. Our main results imply that simplices can only be ``trivially approximated''. On the way, we obtain a characterization of simplices as the only ``outer additive'' convex bodies.

Authors: Egor Bakaev, Florestan Brunck, Amir Yehudayoff

We study approximations of polytopes in the standard model for computing polytopes using Minkowski sums and (convex hulls of) unions. Specifically, we study the ability to approximate a target polytope by polytopes of a given depth. Our main results imply that simplices can only be ``trivially approximated''. On the way, we obtain a characterization of simplices as the only ``outer additive'' convex bodies.

Friday, July 11 2025, 00:00

The Smooth Power of the "Neandertal Method"

Authors: Aaron Montag, Tim Reinhardt, Jürgen Richter-Gebert

We describe an algorithmic method to transform a Euclidean wallpaper pattern into a Circle Limit-style picture \`a la Escher. The design goals for the method are to be mathematically sound, aesthetically pleasing and fast to compute. It turns out that a certain class of conformal maps is particularly well-suited for the problem. Moreover, in our specific application, a very simple method, sometimes jokingly called the "Neandertal Method" for its almost brutal simplicity, proves to be highly efficient, as it can easily be parallelized to be run on the GPU, unlike many other approaches.

Authors: Aaron Montag, Tim Reinhardt, Jürgen Richter-Gebert

We describe an algorithmic method to transform a Euclidean wallpaper pattern into a Circle Limit-style picture \`a la Escher. The design goals for the method are to be mathematically sound, aesthetically pleasing and fast to compute. It turns out that a certain class of conformal maps is particularly well-suited for the problem. Moreover, in our specific application, a very simple method, sometimes jokingly called the "Neandertal Method" for its almost brutal simplicity, proves to be highly efficient, as it can easily be parallelized to be run on the GPU, unlike many other approaches.

Friday, July 11 2025, 00:00

A simple proof of a $(p,2)$-theorem for non-piercing regions

Authors: Chaya Keller, Shakhar Smorodinsky

A family of sets satisfies the $(p,2)$-property if among any $p$ sets in the family, some two intersect. Two recent works used elaborate geometric techniques to show that any family of non-piercing regions in the plane that satisfies the $(p,2)$-property can be pierced by $O(p^9)$ points. In this note we show that even in a much more general setting, piercing by $O(p)$ points can be deduced from known results on hypergraphs with a hereditarily linear Delaunay graph, which include intersection hypergraphs of non-piercing regions.

Authors: Chaya Keller, Shakhar Smorodinsky

A family of sets satisfies the $(p,2)$-property if among any $p$ sets in the family, some two intersect. Two recent works used elaborate geometric techniques to show that any family of non-piercing regions in the plane that satisfies the $(p,2)$-property can be pierced by $O(p^9)$ points. In this note we show that even in a much more general setting, piercing by $O(p)$ points can be deduced from known results on hypergraphs with a hereditarily linear Delaunay graph, which include intersection hypergraphs of non-piercing regions.

Friday, July 11 2025, 00:00

On the Complexity of Hyperpath and Minimal Separator Enumeration in Directed Hypergraphs

Authors: Kazuhiro Kurita, Kevin Mann

In this paper, we address the enumeration of (induced) $s$-$t$ paths and minimal $s$-$t$ separators. These problems are some of the most famous classical enumeration problems that can be solved in polynomial delay by simple backtracking for a (un)directed graph. As a generalization of these problems, we consider the (induced) $s$-$t$ hyperpath and minimal $s$-$t$ separator enumeration in a \emph{directed hypergraph}. We show that extending these classical enumeration problems to directed hypergraphs drastically changes their complexity. More precisely, there are no output-polynomial time algorithms for the enumeration of induced $s$-$t$ hyperpaths and minimal $s$-$t$ separators unless $P = NP$, and if there is an output-polynomial time algorithm for the $s$-$t$ hyperpath enumeration, then the minimal transversal enumeration can be solved in output polynomial time even if a directed hypergraph is $BF$-hypergraph. Since the existence of an output-polynomial time algorithm for the minimal transversal enumeration has remained an open problem for over 45 years, it indicates that the $s$-$t$ hyperpath enumeration for a $BF$-hypergraph is not an easy problem. As a positive result, the $s$-$t$ hyperpath enumeration for a $B$-hypergraph can be solved in polynomial delay by backtracking.

Authors: Kazuhiro Kurita, Kevin Mann

In this paper, we address the enumeration of (induced) $s$-$t$ paths and minimal $s$-$t$ separators. These problems are some of the most famous classical enumeration problems that can be solved in polynomial delay by simple backtracking for a (un)directed graph. As a generalization of these problems, we consider the (induced) $s$-$t$ hyperpath and minimal $s$-$t$ separator enumeration in a \emph{directed hypergraph}. We show that extending these classical enumeration problems to directed hypergraphs drastically changes their complexity. More precisely, there are no output-polynomial time algorithms for the enumeration of induced $s$-$t$ hyperpaths and minimal $s$-$t$ separators unless $P = NP$, and if there is an output-polynomial time algorithm for the $s$-$t$ hyperpath enumeration, then the minimal transversal enumeration can be solved in output polynomial time even if a directed hypergraph is $BF$-hypergraph. Since the existence of an output-polynomial time algorithm for the minimal transversal enumeration has remained an open problem for over 45 years, it indicates that the $s$-$t$ hyperpath enumeration for a $BF$-hypergraph is not an easy problem. As a positive result, the $s$-$t$ hyperpath enumeration for a $B$-hypergraph can be solved in polynomial delay by backtracking.

Friday, July 11 2025, 00:00

Finding One Local Optimum Is Easy -- But What about Two?

Authors: Yasuaki Kobayashi, Kazuhiro Kurita, Yutaro Yamaguchi

The class PLS (Polynomial Local Search) captures the complexity of finding a solution that is locally optimal and has proven to be an important concept in the theory of local search. It has been shown that local search versions of various combinatorial optimization problems, such as Maximum Independent Set and Max Cut, are complete for this class. Such computational intractability typically arises in local search problems allowing arbitrary weights; in contrast, for unweighted problems, locally optimal solutions can be found in polynomial time under standard settings. In this paper, we pursue the complexity of local search problems from a different angle: We show that computing two locally optimal solutions is NP-hard for various natural unweighted local search problems, including Maximum Independent Set, Minimum Dominating Set, Max SAT, and Max Cut. We also discuss several tractable cases for finding two (or more) local optimal solutions.

Authors: Yasuaki Kobayashi, Kazuhiro Kurita, Yutaro Yamaguchi

The class PLS (Polynomial Local Search) captures the complexity of finding a solution that is locally optimal and has proven to be an important concept in the theory of local search. It has been shown that local search versions of various combinatorial optimization problems, such as Maximum Independent Set and Max Cut, are complete for this class. Such computational intractability typically arises in local search problems allowing arbitrary weights; in contrast, for unweighted problems, locally optimal solutions can be found in polynomial time under standard settings. In this paper, we pursue the complexity of local search problems from a different angle: We show that computing two locally optimal solutions is NP-hard for various natural unweighted local search problems, including Maximum Independent Set, Minimum Dominating Set, Max SAT, and Max Cut. We also discuss several tractable cases for finding two (or more) local optimal solutions.

Friday, July 11 2025, 00:00

Finding sparse induced subgraphs on graphs of bounded induced matching treewidth

Authors: Hans L. Bodlaender, Fedor V. Fomin, Tuukka Korhonen

The induced matching width of a tree decomposition of a graph $G$ is the cardinality of a largest induced matching $M$ of $G$, such that there exists a bag that intersects every edge in $M$. The induced matching treewidth of a graph $G$, denoted by $\mathsf{tree-}\mu(G)$, is the minimum induced matching width of a tree decomposition of $G$. The parameter $\mathsf{tree-}\mu$ was introduced by Yolov [SODA '18], who showed that, for example, Maximum-Weight Independent Set can be solved in polynomial-time on graphs of bounded $\mathsf{tree-}\mu$. Lima, Milani\v{c}, Mur\v{s}i\v{c}, Okrasa, Rz\k{a}\.zewski, and \v{S}torgel [ESA '24] conjectured that this algorithm can be generalized to a meta-problem called Maximum-Weight Induced Subgraph of Bounded Treewidth, where we are given a vertex-weighted graph $G$, an integer $w$, and a $\mathsf{CMSO}_2$-sentence $\Phi$, and are asked to find a maximum-weight set $X \subseteq V(G)$ so that $G[X]$ has treewidth at most $w$ and satisfies $\Phi$. They proved the conjecture for some special cases, such as for the problem Maximum-Weight Induced Forest. In this paper, we prove the general case of the conjecture. In particular, we show that Maximum-Weight Induced Subgraph of Bounded Treewidth is polynomial-time solvable when $\mathsf{tree-}\mu(G)$, $w$, and $|\Phi|$ are bounded. The running time of our algorithm for $n$-vertex graphs $G$ with $\mathsf{tree} - \mu(G) \le k$ is $f(k, w, |\Phi|) \cdot n^{O(k w^2)}$ for a computable function $f$.

Authors: Hans L. Bodlaender, Fedor V. Fomin, Tuukka Korhonen

The induced matching width of a tree decomposition of a graph $G$ is the cardinality of a largest induced matching $M$ of $G$, such that there exists a bag that intersects every edge in $M$. The induced matching treewidth of a graph $G$, denoted by $\mathsf{tree-}\mu(G)$, is the minimum induced matching width of a tree decomposition of $G$. The parameter $\mathsf{tree-}\mu$ was introduced by Yolov [SODA '18], who showed that, for example, Maximum-Weight Independent Set can be solved in polynomial-time on graphs of bounded $\mathsf{tree-}\mu$. Lima, Milani\v{c}, Mur\v{s}i\v{c}, Okrasa, Rz\k{a}\.zewski, and \v{S}torgel [ESA '24] conjectured that this algorithm can be generalized to a meta-problem called Maximum-Weight Induced Subgraph of Bounded Treewidth, where we are given a vertex-weighted graph $G$, an integer $w$, and a $\mathsf{CMSO}_2$-sentence $\Phi$, and are asked to find a maximum-weight set $X \subseteq V(G)$ so that $G[X]$ has treewidth at most $w$ and satisfies $\Phi$. They proved the conjecture for some special cases, such as for the problem Maximum-Weight Induced Forest. In this paper, we prove the general case of the conjecture. In particular, we show that Maximum-Weight Induced Subgraph of Bounded Treewidth is polynomial-time solvable when $\mathsf{tree-}\mu(G)$, $w$, and $|\Phi|$ are bounded. The running time of our algorithm for $n$-vertex graphs $G$ with $\mathsf{tree} - \mu(G) \le k$ is $f(k, w, |\Phi|) \cdot n^{O(k w^2)}$ for a computable function $f$.

Friday, July 11 2025, 00:00

A Randomized Rounding Approach for DAG Edge Deletion

Authors: Sina Kalantarzadeh, Nathan Klein, Victor Reis

In the DAG Edge Deletion problem, we are given an edge-weighted directed acyclic graph and a parameter $k$, and the goal is to delete the minimum weight set of edges so that the resulting graph has no paths of length $k$. This problem, which has applications to scheduling, was introduced in 2015 by Kenkre, Pandit, Purohit, and Saket. They gave a $k$-approximation and showed that it is UGC-Hard to approximate better than $\lfloor 0.5k \rfloor$ for any constant $k \ge 4$ using a work of Svensson from 2012. The approximation ratio was improved to $\frac{2}{3}(k+1)$ by Klein and Wexler in 2016. In this work, we introduce a randomized rounding framework based on distributions over vertex labels in $[0,1]$. The most natural distribution is to sample labels independently from the uniform distribution over $[0,1]$. We show this leads to a $(2-\sqrt{2})(k+1) \approx 0.585(k+1)$-approximation. By using a modified (but still independent) label distribution, we obtain a $0.549(k+1)$-approximation for the problem, as well as show that no independent distribution over labels can improve our analysis to below $0.542(k+1)$. Finally, we show a $0.5(k+1)$-approximation for bipartite graphs and for instances with structured LP solutions. Whether this ratio can be obtained in general is open.

Authors: Sina Kalantarzadeh, Nathan Klein, Victor Reis

In the DAG Edge Deletion problem, we are given an edge-weighted directed acyclic graph and a parameter $k$, and the goal is to delete the minimum weight set of edges so that the resulting graph has no paths of length $k$. This problem, which has applications to scheduling, was introduced in 2015 by Kenkre, Pandit, Purohit, and Saket. They gave a $k$-approximation and showed that it is UGC-Hard to approximate better than $\lfloor 0.5k \rfloor$ for any constant $k \ge 4$ using a work of Svensson from 2012. The approximation ratio was improved to $\frac{2}{3}(k+1)$ by Klein and Wexler in 2016. In this work, we introduce a randomized rounding framework based on distributions over vertex labels in $[0,1]$. The most natural distribution is to sample labels independently from the uniform distribution over $[0,1]$. We show this leads to a $(2-\sqrt{2})(k+1) \approx 0.585(k+1)$-approximation. By using a modified (but still independent) label distribution, we obtain a $0.549(k+1)$-approximation for the problem, as well as show that no independent distribution over labels can improve our analysis to below $0.542(k+1)$. Finally, we show a $0.5(k+1)$-approximation for bipartite graphs and for instances with structured LP solutions. Whether this ratio can be obtained in general is open.

Friday, July 11 2025, 00:00

Efficient and Adaptive Estimation of Local Triadic Coefficients

Authors: Ilie Sarpe, Aristides Gionis

Characterizing graph properties is fundamental to the analysis and to our understanding of real-world networked systems. The local clustering coefficient, and the more recently introduced, local closure coefficient, capture powerful properties that are essential in a large number of applications, ranging from graph embeddings to graph partitioning. Such coefficients capture the local density of the neighborhood of each node, considering incident triadic structures and paths of length two. For this reason, we refer to these coefficients collectively as local triadic coefficients. In this work, we consider the novel problem of computing efficiently the average of local triadic coefficients, over a given partition of the nodes of the input graph into a set of disjoint buckets. The average local triadic coefficients of the nodes in each bucket provide a better insight into the interplay of graph structure and the properties of the nodes associated to each bucket. Unfortunately, exact computation, which requires listing all triangles in a graph, is infeasible for large networks. Hence, we focus on obtaining highly-accurate probabilistic estimates. We develop Triad, an adaptive algorithm based on sampling, which can be used to estimate the average local triadic coefficients for a partition of the nodes into buckets. Triad is based on a new class of unbiased estimators, and non-trivial bounds on its sample complexity, enabling the efficient computation of highly accurate estimates. Finally, we show how Triad can be efficiently used in practice on large networks, and we present a case study showing that average local triadic coefficients can capture high-order patterns over collaboration networks.

Authors: Ilie Sarpe, Aristides Gionis

Characterizing graph properties is fundamental to the analysis and to our understanding of real-world networked systems. The local clustering coefficient, and the more recently introduced, local closure coefficient, capture powerful properties that are essential in a large number of applications, ranging from graph embeddings to graph partitioning. Such coefficients capture the local density of the neighborhood of each node, considering incident triadic structures and paths of length two. For this reason, we refer to these coefficients collectively as local triadic coefficients. In this work, we consider the novel problem of computing efficiently the average of local triadic coefficients, over a given partition of the nodes of the input graph into a set of disjoint buckets. The average local triadic coefficients of the nodes in each bucket provide a better insight into the interplay of graph structure and the properties of the nodes associated to each bucket. Unfortunately, exact computation, which requires listing all triangles in a graph, is infeasible for large networks. Hence, we focus on obtaining highly-accurate probabilistic estimates. We develop Triad, an adaptive algorithm based on sampling, which can be used to estimate the average local triadic coefficients for a partition of the nodes into buckets. Triad is based on a new class of unbiased estimators, and non-trivial bounds on its sample complexity, enabling the efficient computation of highly accurate estimates. Finally, we show how Triad can be efficiently used in practice on large networks, and we present a case study showing that average local triadic coefficients can capture high-order patterns over collaboration networks.

Friday, July 11 2025, 00:00

An open mindset

from Ben Recht

The commitments required for fully open source machine learning

Over the 4th of July weekend, Nathan Lambert wrote a thoughtful post on “an American Deepseek Project,” posing the challenge of building a fully open, fully performant “frontier model” in the next two years. Deepseek, in case you’ve already forgotten, is a Chinese company that released a highly performing, open-source large language model chatbot in January that could be trained from scratch for under 10 million dollars. Nathan challenges the US research community to produce a similar open artifact.

Nathan’s post reminded me of what happened the last time we were told that you needed to be a hyperscaling tech company to do machine learning. In 2014, a year after paying 44 million dollars for Alex Krizhevsky, Ilya Sutskever, and Geoff Hinton, Google won the ILSVRC competition with their “GoogLeNet” model, achieving 93% top-5 accuracy.1 While the authors didn’t release the full details of how to train this model, they asserted they used Google’s internal DistBelief system, a distributed computing framework for wrangling thousands of Google CPUs. However, based on a back of the envelope calculation, they claimed “ the GoogLeNet network could be trained to convergence using few high-end GPUs within a week, the main limitation being the memory usage.”

Shortly thereafter, a team at Berkeley sort of reproduced this result, showing that half a day with 128 GPUs could get to 88% top-5 accuracy with the GoogLeNet architecture. I imagine they could have reached 93% had they run for longer. A supercomputer with 128 GPUs was still a serious investment for any academic team, and openly trained ImageNet models still seemed to require infrastructural investments outside the reach of most labs.

In 2015, a team of researchers at Microsoft Research Asia in Beijing won the ILSVRC competition with their architecture called the ResNet. ResNets were interesting because the principles of architecture were easier to understand. The repeated layers in ResNets made it easy to explore and reimplement the architecture.

Soon thereafter, the Dawn project at Stanford opened a competition, called DawnBench, to see who could train ResNets the fastest. In November 2017, their baseline entry used 8 GPUs for 10.5 days to train a ResNet ImageNet classifier with 93% top-5 accuracy. This cost $1100 using Amazon cloud services. Already, this was getting somewhere, but I don’t think the Dawn team anticipated how quickly competition progress would come. By April 2018, that is, in less than six months, competitors had the training time down to 30 minutes on a single TPU core on Google Cloud. That’s nearly a 500x acceleration in 6 months, and a dramatic cost reduction.

I tell this story because I’m optimistic that Nathan’s challenge is feasible. For two decades, Google and its peers have argued that you need scale to do anything. These claims are marketing. They have been proven wrong several times,2 but it required a significant community commitment to do it. What are those commitments for Nathan’s Deepseek Project?

First, it’s elevating old school computational evaluations. During the initial deep learning boom, the machine learning research community decided that worthy common tasks were training time and model accessibility. These evaluations have been missing from the latest “frontier models” arms race. It’s also understandable that the computing for “frontier models” is much larger than it was for ImageNet. The hyperscalers' only advantage is to find a starting point that’s inaccessible to a researcher without an infinite cloud computing budget. The 6 million dollars needed to train DeepSeek is out of reach for almost any machine learning researcher. This means that the initial work on any fully open LLM will require large-scale collaboration or perhaps an industrial benefactor. On the other hand, a 500x improvement in training time gets these models down to costing under twenty thousand dollars, which is the sort of thing you can add as a line item in a grant proposal.

More importantly, I want to emphasize that Nathan’s challenge is for something greater than just GPU golf. When he writes “open model,” he really means open. The Deepseek model is “open weights,” which means you can run and tweak it, but you can’t build it from scratch. This is similar to the openness of Meta’s Llama models. Nathan’s vision calls for open data, training code, logs, and decision making too.

I will never get over how the entire machine learning community is writing papers about systems where they don’t know what the training data is. Yes, the artifacts are interesting, but no, we’re not going to learn anything by fine-tuning RL models on top and running hackneyed 1960s psychology experiments on computers. Certainly, open weight models are preferable to closed weight models. But we shouldn’t have to settle.

Open data poses its own challenges to our community. Though we generally concede that open data has been the primary means of driving machine learning research, parts of the research community are openly hostile to open data. And it’s not exactly who you’d think it would be. Some of the biggest “success stories” of the responsible AI/fairness community are based on finding “dangerous” content in open data sets, resulting in these data sets being removed from the public domain. The Netflix Prize, Tiny Images Dataset, and yes, even ImageNet, have all been either entirely removed, taken down for extended periods, or neutered in some capacity because of highly questionable arguments about “responsibility.”

This push towards prioritizing academic notions of “privacy,” “safety,” and “fairness” over all else has mostly resulted in academic censorship. And yet, while the hyperscalers love to pay lip service to responsible behavior, they don’t stop using this data! Recent lawsuits have revealed that Anthropic scans millions of books and Facebook trains on LibGen. But academic researchers aren’t allowed to use the Tiny Images Dataset. The hyperscalers do whatever they want, in private, locking out competition and strengthening their oligopoly. They know they will find a favorable ruling in Bill Alsup’s court because they have infinite money for their legal teams.

So while it’s undeniably essential to raise issues about the societal impacts of machine learning technology, we have to be careful not to do this in a way that solidifies corporate power. If we want fully open and accessible machine learning models, we need to figure out how to build them while accepting that the internet on which they are trained is littered with endlessly horrific and dangerous content. Nathan’s challenge is thus asking the open research community to engage with uncomfortable trade-offs about what open means. This question is as vital as shaking the can to raise money for NVIDIA chips.

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1

For the purposes of this blog post “top-5 accuracy” is just a number between 0 and 100. Higher is better. Don’t sweat the details of what this means or actually evaluates.

2

Naomi Saphra points me to another example in machine translation.

By Ben Recht

Thursday, July 10 2025, 13:59

TR25-091 | Tree PCPs | Tamer Mour, Alon Rosen, Ron Rothblum

from ECCC Papers

Probabilistically checkable proofs (PCPs) allow encoding a computation so that it can be quickly verified by only reading a few symbols. Inspired by tree codes (Schulman, STOC'93), we propose tree PCPs; these are PCPs that evolve as the computation progresses so that a proof for time $t$ is obtained by appending a short string to the end of the proof for time $t-1$. At any given time, the tree PCP can be locally queried to verify the entire computation so far. We construct tree PCPs for non-deterministic space-$s$ computation, where at time step $t$, the proof only grows by an additional $poly(s,\log(t))$ bits, and the number of queries made by the verifier to the overall proof is $poly(s) \cdot t^\epsilon$, for an arbitrary constant $\epsilon > 0$. Tree PCPs are well-suited to proving correctness of ongoing computation that unfolds over time. They may be thought of as an information-theoretic analog of the cryptographic notion of incrementally verifiable computation (Valiant, TCC'08). In the random oracle model, tree PCPs can be compiled to realize a variant of incrementally verifiable computation where the prover is allowed a small number of queries to a large evolving state. This yields the first construction of (a natural variant of) IVC in the random oracle model.

Thursday, July 10 2025, 12:25

TR25-090 | Linear Prover IOPs in Log Star Rounds | Ron Rothblum, Noor Athamnah, Noga Ron-Zewi

from ECCC Papers

Interactive Oracle Proofs (IOPs) form the backbone of some of the most efficient general-purpose cryptographic proof-systems. In an IOP, the prover can interact with the verifier over multiple rounds, where in each round the prover sends a long message, from which the verifier only queries a few symbols. State-of-the-art IOPs achieve a linear-size prover and a poly-logarithmic verifier but require a relatively large, logarithmic, number of rounds. While Fiat-Shamir heuristic can be used to eliminate the need for actual interaction, in modern highly-parallelizable computer architectures such as GPUs, the large number of rounds still translates into a major bottleneck for the prover, since it needs to alternate between computing the IOP messages and the Fiat-Shamir hashes. Motivated by this fact, in this work we study the round complexity of linear-prover IOPs. Our main result is an IOP for a large class of Boolean circuits, with only $O(\log^*(S))$ rounds, where $\log^*$ denotes the iterated logarithm function (and $S$ is the circuit size). The prover has linear size $O(S)$ and the verifier runs in time $\mathrm{polylog}(S)$ and has query complexity $O(\log^*(S))$. The protocol is both conceptually simpler, and strictly more efficient, than prior linear prover IOPs for Boolean circuits.

Thursday, July 10 2025, 12:23

Efficient Algorithms for Quantum Hashing

Authors: Ilnar Zinnatullin, Kamil Khadiev

Quantum hashing is a useful technique that allows us to construct memory-efficient algorithms and secure quantum protocols. First, we present a circuit that implements the phase form of quantum hashing using $2^{n-1}$ CNOT gates, where n is the number of control qubits. Our method outperforms existing approaches and reduces the circuit depth. Second, we propose an algorithm that provides a trade-off between the number of CNOT gates (and consequently, the circuit depth) and the precision of rotation angles. This is particularly important in the context of NISQ (Noisy Intermediate-Scale Quantum) devices, where hardware-imposed angle precision limit remains a critical constraint.

Authors: Ilnar Zinnatullin, Kamil Khadiev

Quantum hashing is a useful technique that allows us to construct memory-efficient algorithms and secure quantum protocols. First, we present a circuit that implements the phase form of quantum hashing using $2^{n-1}$ CNOT gates, where n is the number of control qubits. Our method outperforms existing approaches and reduces the circuit depth. Second, we propose an algorithm that provides a trade-off between the number of CNOT gates (and consequently, the circuit depth) and the precision of rotation angles. This is particularly important in the context of NISQ (Noisy Intermediate-Scale Quantum) devices, where hardware-imposed angle precision limit remains a critical constraint.

Thursday, July 10 2025, 00:00

Trainability of Quantum Models Beyond Known Classical Simulability

Authors: Sabri Meyer, Francesco Scala, Francesco Tacchino, Aurelien Lucchi

Variational Quantum Algorithms (VQAs) are promising candidates for near-term quantum computing, yet they face scalability challenges due to barren plateaus, where gradients vanish exponentially in the system size. Recent conjectures suggest that avoiding barren plateaus might inherently lead to classical simulability, thus limiting the opportunities for quantum advantage. In this work, we advance the theoretical understanding of the relationship between the trainability and computational complexity of VQAs, thus directly addressing the conjecture. We introduce the Linear Clifford Encoder (LCE), a novel technique that ensures constant-scaling gradient statistics on optimization landscape regions that are close to Clifford circuits. Additionally, we leverage classical Taylor surrogates to reveal computational complexity phase transitions from polynomial to super-polynomial as the initialization region size increases. Combining these results, we reveal a deeper link between trainability and computational complexity, and analytically prove that barren plateaus can be avoided in regions for which no classical surrogate is known to exist. Furthermore, numerical experiments on LCE transformed landscapes confirm in practice the existence of a super-polynomially complex ``transition zone'' where gradients decay polynomially. These findings indicate a plausible path to practically relevant, barren plateau-free variational models with potential for quantum advantage.

Authors: Sabri Meyer, Francesco Scala, Francesco Tacchino, Aurelien Lucchi

Variational Quantum Algorithms (VQAs) are promising candidates for near-term quantum computing, yet they face scalability challenges due to barren plateaus, where gradients vanish exponentially in the system size. Recent conjectures suggest that avoiding barren plateaus might inherently lead to classical simulability, thus limiting the opportunities for quantum advantage. In this work, we advance the theoretical understanding of the relationship between the trainability and computational complexity of VQAs, thus directly addressing the conjecture. We introduce the Linear Clifford Encoder (LCE), a novel technique that ensures constant-scaling gradient statistics on optimization landscape regions that are close to Clifford circuits. Additionally, we leverage classical Taylor surrogates to reveal computational complexity phase transitions from polynomial to super-polynomial as the initialization region size increases. Combining these results, we reveal a deeper link between trainability and computational complexity, and analytically prove that barren plateaus can be avoided in regions for which no classical surrogate is known to exist. Furthermore, numerical experiments on LCE transformed landscapes confirm in practice the existence of a super-polynomially complex ``transition zone'' where gradients decay polynomially. These findings indicate a plausible path to practically relevant, barren plateau-free variational models with potential for quantum advantage.

Thursday, July 10 2025, 00:00

An Improved Bound for Plane Covering Paths

Authors: Hugo A. Akitaya, Greg Aloupis, Ahmad Biniaz, Prosenjit Bose, Jean-Lou De Carufel, Cyril Gavoille, John Iacono, Linda Kleist, Michiel Smid, Diane Souvaine, Leonidas Theocharous

A covering path for a finite set $P$ of points in the plane is a polygonal path such that every point of $P$ lies on a segment of the path. The vertices of the path need not be at points of $P$. A covering path is plane if its segments do not cross each other. Let $\pi(n)$ be the minimum number such that every set of $n$ points in the plane admits a plane covering path with at most $\pi(n)$ segments. We prove that $\pi(n)\le \lceil6n/7\rceil$. This improves the previous best-known upper bound of $\lceil 21n/22\rceil$, due to Biniaz (SoCG 2023). Our proof is constructive and yields a simple $O(n\log n)$-time algorithm for computing a plane covering path.

Authors: Hugo A. Akitaya, Greg Aloupis, Ahmad Biniaz, Prosenjit Bose, Jean-Lou De Carufel, Cyril Gavoille, John Iacono, Linda Kleist, Michiel Smid, Diane Souvaine, Leonidas Theocharous

A covering path for a finite set $P$ of points in the plane is a polygonal path such that every point of $P$ lies on a segment of the path. The vertices of the path need not be at points of $P$. A covering path is plane if its segments do not cross each other. Let $\pi(n)$ be the minimum number such that every set of $n$ points in the plane admits a plane covering path with at most $\pi(n)$ segments. We prove that $\pi(n)\le \lceil6n/7\rceil$. This improves the previous best-known upper bound of $\lceil 21n/22\rceil$, due to Biniaz (SoCG 2023). Our proof is constructive and yields a simple $O(n\log n)$-time algorithm for computing a plane covering path.

Thursday, July 10 2025, 00:00

Topological Machine Learning with Unreduced Persistence Diagrams

Authors: Nicole Abreu, Parker B. Edwards, Francis Motta

Supervised machine learning pipelines trained on features derived from persistent homology have been experimentally observed to ignore much of the information contained in a persistence diagram. Computing persistence diagrams is often the most computationally demanding step in such a pipeline, however. To explore this, we introduce several methods to generate topological feature vectors from unreduced boundary matrices. We compared the performance of pipelines trained on vectorizations of unreduced PDs to vectorizations of fully-reduced PDs across several data and task types. Our results indicate that models trained on PDs built from unreduced diagrams can perform on par and even outperform those trained on fully-reduced diagrams on some tasks. This observation suggests that machine learning pipelines which incorporate topology-based features may benefit in terms of computational cost and performance by utilizing information contained in unreduced boundary matrices.

Authors: Nicole Abreu, Parker B. Edwards, Francis Motta

Supervised machine learning pipelines trained on features derived from persistent homology have been experimentally observed to ignore much of the information contained in a persistence diagram. Computing persistence diagrams is often the most computationally demanding step in such a pipeline, however. To explore this, we introduce several methods to generate topological feature vectors from unreduced boundary matrices. We compared the performance of pipelines trained on vectorizations of unreduced PDs to vectorizations of fully-reduced PDs across several data and task types. Our results indicate that models trained on PDs built from unreduced diagrams can perform on par and even outperform those trained on fully-reduced diagrams on some tasks. This observation suggests that machine learning pipelines which incorporate topology-based features may benefit in terms of computational cost and performance by utilizing information contained in unreduced boundary matrices.

Thursday, July 10 2025, 00:00

Faster Estimation of the Average Degree of a Graph Using Random Edges and Structural Queries

Authors: Lorenzo Beretta, Deeparnab Chakrabarty, C. Seshadhri

We revisit the problem of designing sublinear algorithms for estimating the average degree of an $n$-vertex graph. The standard access model for graphs allows for the following queries: sampling a uniform random vertex, the degree of a vertex, sampling a uniform random neighbor of a vertex, and ``pair queries'' which determine if a pair of vertices form an edge. In this model, original results [Goldreich-Ron, RSA 2008; Eden-Ron-Seshadhri, SIDMA 2019] on this problem prove that the complexity of getting $(1+\varepsilon)$-multiplicative approximations to the average degree, ignoring $\varepsilon$-dependencies, is $\Theta(\sqrt{n})$. When random edges can be sampled, it is known that the average degree can estimated in $\widetilde{O}(n^{1/3})$ queries, even without pair queries [Motwani-Panigrahy-Xu, ICALP 2007; Beretta-Tetek, TALG 2024]. We give a nearly optimal algorithm in the standard access model with random edge samples. Our algorithm makes $\widetilde{O}(n^{1/4})$ queries exploiting the power of pair queries. We also analyze the ``full neighborhood access" model wherein the entire adjacency list of a vertex can be obtained with a single query; this model is relevant in many practical applications. In a weaker version of this model, we give an algorithm that makes $\widetilde{O}(n^{1/5})$ queries. Both these results underscore the power of {\em structural queries}, such as pair queries and full neighborhood access queries, for estimating the average degree. We give nearly matching lower bounds, ignoring $\varepsilon$-dependencies, for all our results. So far, almost all algorithms for estimating average degree assume that the number of vertices, $n$, is known. Inspired by [Beretta-Tetek, TALG 2024], we study this problem when $n$ is unknown and show that structural queries do not help in estimating average degree in this setting.

Authors: Lorenzo Beretta, Deeparnab Chakrabarty, C. Seshadhri

We revisit the problem of designing sublinear algorithms for estimating the average degree of an $n$-vertex graph. The standard access model for graphs allows for the following queries: sampling a uniform random vertex, the degree of a vertex, sampling a uniform random neighbor of a vertex, and ``pair queries'' which determine if a pair of vertices form an edge. In this model, original results [Goldreich-Ron, RSA 2008; Eden-Ron-Seshadhri, SIDMA 2019] on this problem prove that the complexity of getting $(1+\varepsilon)$-multiplicative approximations to the average degree, ignoring $\varepsilon$-dependencies, is $\Theta(\sqrt{n})$. When random edges can be sampled, it is known that the average degree can estimated in $\widetilde{O}(n^{1/3})$ queries, even without pair queries [Motwani-Panigrahy-Xu, ICALP 2007; Beretta-Tetek, TALG 2024]. We give a nearly optimal algorithm in the standard access model with random edge samples. Our algorithm makes $\widetilde{O}(n^{1/4})$ queries exploiting the power of pair queries. We also analyze the ``full neighborhood access" model wherein the entire adjacency list of a vertex can be obtained with a single query; this model is relevant in many practical applications. In a weaker version of this model, we give an algorithm that makes $\widetilde{O}(n^{1/5})$ queries. Both these results underscore the power of {\em structural queries}, such as pair queries and full neighborhood access queries, for estimating the average degree. We give nearly matching lower bounds, ignoring $\varepsilon$-dependencies, for all our results. So far, almost all algorithms for estimating average degree assume that the number of vertices, $n$, is known. Inspired by [Beretta-Tetek, TALG 2024], we study this problem when $n$ is unknown and show that structural queries do not help in estimating average degree in this setting.

Thursday, July 10 2025, 00:00

Faster Algorithms for $(2k-1)$-Stretch Distance Oracles

Authors: Avi Kadria, Liam Roditty

Let $G=(V, E)$ be an undirected $n$-vertices $m$-edges graph with non-negative edge weights. In this paper, we present three new algorithms for constructing a $(2k-1)$-stretch distance oracle with $O(n^{1+\frac{1}{k}})$ space. The first algorithm runs in $\Ot(\max(n^{1+2/k}, m^{1-\frac{1}{k-1}}n^{\frac{2}{k-1}}))$ time, and improves upon the $\Ot(\min(mn^{\frac{1}{k}},n^2))$ time of Thorup and Zwick [STOC 2001, JACM 2005] and Baswana and Kavitha [FOCS 2006, SICOMP 2010], for every $k > 2$ and $m=\Omega(n^{1+\frac{1}{k}+\eps})$. This yields the first truly subquadratic time construction for every $2 < k < 6$, and nearly resolves the open problem posed by Wulff-Nilsen [SODA 2012] on the existence of such constructions. The two other algorithms have a running time of the form $\Ot(m+n^{1+f(k)})$, which is near linear in $m$ if $m=\Omega(n^{1+f(k)})$, and therefore optimal in such graphs. One algorithm runs in $\Ot(m+n^{\frac32+\frac{3}{4k-6}})$-time, which improves upon the $\Ot(n^2)$-time algorithm of Baswana and Kavitha [FOCS 2006, SICOMP 2010], for $3 < k < 6$, and upon the $\Ot(m+n^{\frac{3}{2}+\frac{2}{k}+O(k^{-2})})$-time algorithm of Wulff-Nilsen [SODA 2012], for every $k\geq 6$. This is the first linear time algorithm for constructing a $7$-stretch distance oracle and a $9$-stretch distance oracle, for graphs with truly subquadratic density.\footnote{with $m=n^{2-\eps}$ for some $\eps > 0$.} The other algorithm runs in $\Ot(\sqrt{k}m+kn^{1+\frac{2\sqrt{2}}{\sqrt{k}}})$ time, (and hence relevant only for $k\ge 16$), and improves upon the $\Ot(\sqrt{k}m+kn^{1+\frac{2\sqrt{6}}{\sqrt{k}}+O(k^{-1})})$ time algorithm of Wulff-Nilsen [SODA 2012] (which is relevant only for $k\ge 96$). ...

Authors: Avi Kadria, Liam Roditty

Let $G=(V, E)$ be an undirected $n$-vertices $m$-edges graph with non-negative edge weights. In this paper, we present three new algorithms for constructing a $(2k-1)$-stretch distance oracle with $O(n^{1+\frac{1}{k}})$ space. The first algorithm runs in $\Ot(\max(n^{1+2/k}, m^{1-\frac{1}{k-1}}n^{\frac{2}{k-1}}))$ time, and improves upon the $\Ot(\min(mn^{\frac{1}{k}},n^2))$ time of Thorup and Zwick [STOC 2001, JACM 2005] and Baswana and Kavitha [FOCS 2006, SICOMP 2010], for every $k > 2$ and $m=\Omega(n^{1+\frac{1}{k}+\eps})$. This yields the first truly subquadratic time construction for every $2 < k < 6$, and nearly resolves the open problem posed by Wulff-Nilsen [SODA 2012] on the existence of such constructions. The two other algorithms have a running time of the form $\Ot(m+n^{1+f(k)})$, which is near linear in $m$ if $m=\Omega(n^{1+f(k)})$, and therefore optimal in such graphs. One algorithm runs in $\Ot(m+n^{\frac32+\frac{3}{4k-6}})$-time, which improves upon the $\Ot(n^2)$-time algorithm of Baswana and Kavitha [FOCS 2006, SICOMP 2010], for $3 < k < 6$, and upon the $\Ot(m+n^{\frac{3}{2}+\frac{2}{k}+O(k^{-2})})$-time algorithm of Wulff-Nilsen [SODA 2012], for every $k\geq 6$. This is the first linear time algorithm for constructing a $7$-stretch distance oracle and a $9$-stretch distance oracle, for graphs with truly subquadratic density.\footnote{with $m=n^{2-\eps}$ for some $\eps > 0$.} The other algorithm runs in $\Ot(\sqrt{k}m+kn^{1+\frac{2\sqrt{2}}{\sqrt{k}}})$ time, (and hence relevant only for $k\ge 16$), and improves upon the $\Ot(\sqrt{k}m+kn^{1+\frac{2\sqrt{6}}{\sqrt{k}}+O(k^{-1})})$ time algorithm of Wulff-Nilsen [SODA 2012] (which is relevant only for $k\ge 96$). ...

Thursday, July 10 2025, 00:00

Designing Parallel Algorithms for Community Detection using Arachne

Authors: Fuhuan Li, Zhihui Du, David A. Bader

The rise of graph data in various fields calls for efficient and scalable community detection algorithms. In this paper, we present parallel implementations of two widely used algorithms: Label Propagation and Louvain, specifically designed to leverage the capabilities of Arachne which is a Python-accessible, open-source framework for large-scale graph analysis. Our implementations achieve substantial speedups over existing Python-based tools like NetworkX and igraph, which lack efficient parallelization, and are competitive with parallel frameworks such as NetworKit. Experimental results show that Arachne-based methods outperform these baselines, achieving speedups of up to 710x over NetworkX, 75x over igraph, and 12x over NetworKit. Additionally, we analyze the scalability of our implementation under varying thread counts, demonstrating how different phases contribute to overall performance gains of the parallel Louvain algorithm. Arachne, including our community detection implementation, is open-source and available at github.com/Bears-R-Us/arkouda-njit .

Authors: Fuhuan Li, Zhihui Du, David A. Bader

The rise of graph data in various fields calls for efficient and scalable community detection algorithms. In this paper, we present parallel implementations of two widely used algorithms: Label Propagation and Louvain, specifically designed to leverage the capabilities of Arachne which is a Python-accessible, open-source framework for large-scale graph analysis. Our implementations achieve substantial speedups over existing Python-based tools like NetworkX and igraph, which lack efficient parallelization, and are competitive with parallel frameworks such as NetworKit. Experimental results show that Arachne-based methods outperform these baselines, achieving speedups of up to 710x over NetworkX, 75x over igraph, and 12x over NetworKit. Additionally, we analyze the scalability of our implementation under varying thread counts, demonstrating how different phases contribute to overall performance gains of the parallel Louvain algorithm. Arachne, including our community detection implementation, is open-source and available at https://github.com/Bears-R-Us/arkouda-njit .

Thursday, July 10 2025, 00:00

Multi-Queue SSD I/O Modeling & Its Implications for Data Structure Design

Authors: Erin Ransom, Andrew Lim, Michael Mitzenmacher

Understanding the performance profiles of storage devices and how best to utilize them has always been non-trivial due to factors such as seek times, caching, scheduling, concurrent access, flash wear-out, and garbage collection. However, analytical frameworks that provide simplified abstractions of storage performance can still be accurate enough to evaluate external memory algorithms and data structures at the design stage. For example, the Disk Access Machine (DAM) model assumes that a storage device transfers data in fixed-size blocks of size B and that all transfers have unit latency. This abstraction is already sufficient to explain some of the benefits of data structures such as B-trees and Log-Structured Merge trees (LSM trees); however, storage technology advances have significantly reduced current models' accuracy and utility. This paper introduces the Multi-Queue Solid State Drive (MQSSD) model, a new storage abstraction. This model builds upon previous models and aims to more accurately represent the performance characteristics of modern storage hardware. We identify key performance-critical aspects of modern multi-queue solid-state drives on which we base our model and demonstrate these characteristics on actual hardware. We then show how our model can be applied to LSM-tree-based storage engines to optimize them for modern storage hardware. We highlight that leveraging concurrent access is crucial for fully utilizing the high throughput of multi-queue SSDs, enabling designs that may appear counterintuitive under traditional paradigms We then validate these insights through experiments using Facebook's LSM-tree-based key-value store, RocksDB. We conclude that the MQSSD model offers a more accurate abstraction of modern hardware than previous models, allowing for greater insight and optimization.

Authors: Erin Ransom, Andrew Lim, Michael Mitzenmacher

Understanding the performance profiles of storage devices and how best to utilize them has always been non-trivial due to factors such as seek times, caching, scheduling, concurrent access, flash wear-out, and garbage collection. However, analytical frameworks that provide simplified abstractions of storage performance can still be accurate enough to evaluate external memory algorithms and data structures at the design stage. For example, the Disk Access Machine (DAM) model assumes that a storage device transfers data in fixed-size blocks of size B and that all transfers have unit latency. This abstraction is already sufficient to explain some of the benefits of data structures such as B-trees and Log-Structured Merge trees (LSM trees); however, storage technology advances have significantly reduced current models' accuracy and utility. This paper introduces the Multi-Queue Solid State Drive (MQSSD) model, a new storage abstraction. This model builds upon previous models and aims to more accurately represent the performance characteristics of modern storage hardware. We identify key performance-critical aspects of modern multi-queue solid-state drives on which we base our model and demonstrate these characteristics on actual hardware. We then show how our model can be applied to LSM-tree-based storage engines to optimize them for modern storage hardware. We highlight that leveraging concurrent access is crucial for fully utilizing the high throughput of multi-queue SSDs, enabling designs that may appear counterintuitive under traditional paradigms We then validate these insights through experiments using Facebook's LSM-tree-based key-value store, RocksDB. We conclude that the MQSSD model offers a more accurate abstraction of modern hardware than previous models, allowing for greater insight and optimization.

Thursday, July 10 2025, 00:00

Parallel Batch-Dynamic Algorithms for Spanners, and Extensions

Authors: Mohsen Ghaffari, Jaehyun Koo

This paper presents the first parallel batch-dynamic algorithms for computing spanners and sparsifiers. Our algorithms process any batch of edge insertions and deletions in an $n$-node undirected graph, in $\text{poly}(\log n)$ depth and using amortized work near-linear in the batch size. Our concrete results are as follows: - Our base algorithm maintains a spanner with $(2k-1)$ stretch and $\tilde{O}(n^{1+1/k})$ edges, for any $k\geq 1$. - Our first extension maintains a sparse spanner with only $O(n)$ edges, and $\tilde{O}(\log n)$ stretch. - Our second extension maintains a $t$-bundle of spanners -- i.e., $t$ spanners, each of which is the spanner of the graph remaining after removing the previous ones -- and allows us to maintain cut/spectral sparsifiers with $\tilde{O}(n)$ edges.

Authors: Mohsen Ghaffari, Jaehyun Koo

This paper presents the first parallel batch-dynamic algorithms for computing spanners and sparsifiers. Our algorithms process any batch of edge insertions and deletions in an $n$-node undirected graph, in $\text{poly}(\log n)$ depth and using amortized work near-linear in the batch size. Our concrete results are as follows: - Our base algorithm maintains a spanner with $(2k-1)$ stretch and $\tilde{O}(n^{1+1/k})$ edges, for any $k\geq 1$. - Our first extension maintains a sparse spanner with only $O(n)$ edges, and $\tilde{O}(\log n)$ stretch. - Our second extension maintains a $t$-bundle of spanners -- i.e., $t$ spanners, each of which is the spanner of the graph remaining after removing the previous ones -- and allows us to maintain cut/spectral sparsifiers with $\tilde{O}(n)$ edges.

Thursday, July 10 2025, 00:00

Parallel Batch-Dynamic Coreness Decomposition with Worst-Case Guarantees

Authors: Mohsen Ghaffari, Jaehyun Koo

We present the first parallel batch-dynamic algorithm for approximating coreness decomposition with worst-case update times. Given any batch of edge insertions and deletions, our algorithm processes all these updates in $ \text{poly}(\log n)$ depth, using a worst-case work bound of $b\cdot \text{poly}(\log n)$ where $b$ denotes the batch size. This means the batch gets processed in $\tilde{O}(b/p)$ time, given $p$ processors, which is optimal up to logarithmic factors. Previously, an algorithm with similar guarantees was known by the celebrated work of Liu, Shi, Yu, Dhulipala, and Shun [SPAA'22], but with the caveat of the work bound, and thus the runtime, being only amortized.

Authors: Mohsen Ghaffari, Jaehyun Koo

We present the first parallel batch-dynamic algorithm for approximating coreness decomposition with worst-case update times. Given any batch of edge insertions and deletions, our algorithm processes all these updates in $ \text{poly}(\log n)$ depth, using a worst-case work bound of $b\cdot \text{poly}(\log n)$ where $b$ denotes the batch size. This means the batch gets processed in $\tilde{O}(b/p)$ time, given $p$ processors, which is optimal up to logarithmic factors. Previously, an algorithm with similar guarantees was known by the celebrated work of Liu, Shi, Yu, Dhulipala, and Shun [SPAA'22], but with the caveat of the work bound, and thus the runtime, being only amortized.

Thursday, July 10 2025, 00:00

TR25-089 | Chain Rules for Time-Bounded Kolmogorov Complexity | Valentine Kabanets, Antonina Kolokolova

from ECCC Papers

Time-bounded conditional Kolmogorov complexity of a string $x$ given $y$, $K^t(x\mid y)$, is the length of a shortest program that, given $y$, prints $x$ within $t$ steps. The Chain Rule for conditional $K^t$ with error $e$ is the following hypothesis: there is a constant $c\in\mathbb{N}$ such that, for any strings $y,x_1,\dots,x_{\ell}\in\{0,1\}^*$, for any $\ell\in\mathbb{N}$, and all sufficiently large time bounds $t$, \[ K^t(x_1,\dots,x_{\ell}\mid y) \geq \sum_{i=1}^{\ell} K^{t^{c}}(x_i \mid y, x_1,\dots,x_{i-1}) - \ell\cdot O(\log t) -e(N,t), \] where $N=\sum_{i=1}^{\ell} |x_i|$. We pinpoint the complexity assumptions equivalent to Chain Rules for conditional $K^t$, and the probabilistic variant $pK^t$, where $pK^t(x\mid y)\leq s$ iff $K^t(x\mid y,r)\leq s$ for at least $2/3$ of random strings $r\in\{0,1\}^t$. 1. Chain Rule for conditional $K^t$ with error $e(N,t)\leq o(N)$ is equivalent to the conjunction of the following two statements: (a) $E\not\subset io SIZE[2^{o(n)}]$, and (b) $Gap McK^tP\in promise\text{-} P$, where $Gap McK^tP$ is a promise problem to distinguish between inputs $(x,y,1^s)$ with $K^t(x\mid y)\leq s$ and those with $K^{poly(t)}(x\mid y)> s + o(|x|)$. 2. Chain Rule for conditional $pK^t$ with error $e(N,t)\leq o(N)$ is equivalent to $Gap McpK^tP\in promise\text{-} BPP$, for the analog of $Gap McK^tP$ for conditional $pK^t$. These are the first exact complexity characterizations for natural versions of Chain Rules for time-bounded Kolmogorov complexity. Assuming $Gap McK^tP$ is $NP$-hard (which is true under cryptographic assumptions [Huang et al., STOC'23]), the equivalence above would simplify to ``the Chain Rule for conditional $K^t$ with error $e(N,t)\leq o(N)$ holds iff $NP=P$''. That is, under a plausible $NP$-hardness assumption for $Gap McK^tP$, we would get that proving $P\neq NP$ is equivalent to disproving the Chain Rule for conditional $K^t$. Among other results, we present a natural $promise\text{-} BPP$-complete problem based on the problem of approximating $pK^t(x\mid y)$ for short inputs $x$ with $|x|\leq \log t$, and give some algorithmic consequences if $Gap McpK^TP$ were easy.

Wednesday, July 09 2025, 22:47

The Customers of the Academy

from Computational Complexity

I had an epiphany reading an article in the Trenton Times when I lived in New Jersey at the turn of the century. The article interviewed companies along a certain street lobbying for a new bus route so their employees could more easily get to work. The customers for mass transit are not its riders but the employers who need workers to get to them. Maybe that's why mass transit trains and buses offer a functional but not particularly comfortable ride.

So who are the customers for universities? Before I go there, let's look at newspapers. Until the early 2000s, newspapers were primarily driven by advertising revenue. Readers were the product. While newspapers needed readers to sell, they could get them by offering cheap subscriptions by focusing on quality coverage that focused on news and analysis from a broad range of views. But since then, the few newspapers that thrive now do so mostly on subscription revenue, print and digital, and the readers have become the customers. They also have more competition from other sources like social media. So newspapers now tailor their coverage and their brand for the paying subscriber, and while most still focus on accuracy, they'll stick to narrower views in their analysis which often overshadows the pure news.
Universities have a mission beyond just serving students, providing them with knowledge in exchange for tuition. They have a societal mission. The Morrill Land-Grant Act of 1862, which helped establish and grow a number of public universities, wanted to educate students to improve the productivity of American agriculture and industry. The GI Bill in 1944 brought the masses of returning soldiers into higher education. The Higher Education Act of 1965, brought in resources for students through Pell Grants and federally-guaranteed student Loans to further the competitiveness of America through the Cold War. Most universities have non-profit status because of their broader mission.
In other words, society as a whole was our customer. Our role is to educate and prepare students to help push our society forward. Many universities also have a research mission, also mostly government funded, both to recruit expert professors to educate our students, but also to produce important knowledge to manage the complexities of the world. Students participated willingly for future intellectual and financial gain and our role was to ensure the students got a strong education, for the betterment of not just themselves but the workforce and society they would later join.
Our viewpoint has changed as college costs increased and universities became more dependent on tuition and governmental financial aid. Institutions started treating the students as the customer, to ensure they came to the university and stayed there. More amenities, grade inflation, much more student support and tolerance. The relationship became transactional, a student comes, pays their tuition and their time, gets a degree and gets a job. The focus becomes more on degrees that prepare you for the workplace, a focus more on immediate skill and credential building than producing students who have the critical thinking skills to build a strong career.
And now in a time of changing demographics, less government support and AI heading towards performing many of the skills universities teach, how does the story continue? How do universities focus back on producing students who can not just live in our society but improve it? How do they focus on the right customers while ensuring educational quality? Universities need to get it right, or they won't have customers at all.

By Lance Fortnow

I had an epiphany reading an article in the Trenton Times when I lived in New Jersey at the turn of the century. The article interviewed companies along a certain street lobbying for a new bus route so their employees could more easily get to work. The customers for mass transit are not its riders but the employers who need workers to get to them. Maybe that's why mass transit trains and buses offer a functional but not particularly comfortable ride.

So who are the customers for universities? Before I go there, let's look at newspapers. Until the early 2000s, newspapers were primarily driven by advertising revenue. Readers were the product. While newspapers needed readers to sell, they could get them by offering cheap subscriptions by focusing on quality coverage that focused on news and analysis from a broad range of views. But since then, the few newspapers that thrive now do so mostly on subscription revenue, print and digital, and the readers have become the customers. They also have more competition from other sources like social media. So newspapers now tailor their coverage and their brand for the paying subscriber, and while most still focus on accuracy, they'll stick to narrower views in their analysis which often overshadows the pure news.

Universities have a mission beyond just serving students, providing them with knowledge in exchange for tuition. They have a societal mission. The Morrill Land-Grant Act of 1862, which helped establish and grow a number of public universities, wanted to educate students to improve the productivity of American agriculture and industry. The GI Bill in 1944 brought the masses of returning soldiers into higher education. The Higher Education Act of 1965, brought in resources for students through Pell Grants and federally-guaranteed student Loans to further the competitiveness of America through the Cold War. Most universities have non-profit status because of their broader mission.

In other words, society as a whole was our customer. Our role is to educate and prepare students to help push our society forward. Many universities also have a research mission, also mostly government funded, both to recruit expert professors to educate our students, but also to produce important knowledge to manage the complexities of the world. Students participated willingly for future intellectual and financial gain and our role was to ensure the students got a strong education, for the betterment of not just themselves but the workforce and society they would later join.

Our viewpoint has changed as college costs increased and universities became more dependent on tuition and governmental financial aid. Institutions started treating the students as the customer, to ensure they came to the university and stayed there. More amenities, grade inflation, much more student support and tolerance. The relationship became transactional, a student comes, pays their tuition and their time, gets a degree and gets a job. The focus becomes more on degrees that prepare you for the workplace, a focus more on immediate skill and credential building than producing students who have the critical thinking skills to build a strong career.

And now in a time of changing demographics, less government support and AI heading towards performing many of the skills universities teach, how does the story continue? How do universities focus back on producing students who can not just live in our society but improve it? How do they focus on the right customers while ensuring educational quality? Universities need to get it right, or they won't have customers at all.

By Lance Fortnow

Wednesday, July 09 2025, 20:20

Unitary designs in nearly optimal depth

Authors: Laura Cui, Thomas Schuster, Fernando Brandao, Hsin-Yuan Huang

We construct $\varepsilon$-approximate unitary $k$-designs on $n$ qubits in circuit depth $O(\log k \log \log n k / \varepsilon)$. The depth is exponentially improved over all known results in all three parameters $n$, $k$, $\varepsilon$. We further show that each dependence is optimal up to exponentially smaller factors. Our construction uses $\tilde{{O}}(nk)$ ancilla qubits and ${O}(nk)$ bits of randomness, which are also optimal up to $\log(n k)$ factors. An alternative construction achieves a smaller ancilla count $\tilde{{O}}(n)$ with circuit depth ${O}(k \log \log nk/\varepsilon)$. To achieve these efficient unitary designs, we introduce a highly-structured random unitary ensemble that leverages long-range two-qubit gates and low-depth implementations of random classical hash functions. We also develop a new analytical framework for bounding errors in quantum experiments involving many queries to random unitaries. As an illustration of this framework's versatility, we provide a succinct alternative proof of the existence of pseudorandom unitaries.

Authors: Laura Cui, Thomas Schuster, Fernando Brandao, Hsin-Yuan Huang

We construct $\varepsilon$-approximate unitary $k$-designs on $n$ qubits in circuit depth $O(\log k \log \log n k / \varepsilon)$. The depth is exponentially improved over all known results in all three parameters $n$, $k$, $\varepsilon$. We further show that each dependence is optimal up to exponentially smaller factors. Our construction uses $\tilde{{O}}(nk)$ ancilla qubits and ${O}(nk)$ bits of randomness, which are also optimal up to $\log(n k)$ factors. An alternative construction achieves a smaller ancilla count $\tilde{{O}}(n)$ with circuit depth ${O}(k \log \log nk/\varepsilon)$. To achieve these efficient unitary designs, we introduce a highly-structured random unitary ensemble that leverages long-range two-qubit gates and low-depth implementations of random classical hash functions. We also develop a new analytical framework for bounding errors in quantum experiments involving many queries to random unitaries. As an illustration of this framework's versatility, we provide a succinct alternative proof of the existence of pseudorandom unitaries.

Wednesday, July 09 2025, 00:00

Generalized and Unified Equivalences between Hardness and Pseudoentropy

Authors: Lunjia Hu, Salil Vadhan

Pseudoentropy characterizations provide a quantitatively precise demonstration of the close relationship between computational hardness and computational randomness. We prove a unified pseudoentropy characterization that generalizes and strengthens previous results for both uniform and non-uniform models of computation. Our characterization holds for a general family of entropy notions that encompasses the common notions of Shannon entropy and min entropy as special cases. Moreover, we show that the characterizations for different entropy notions can be simultaneously achieved by a single, universal function that simultaneously witnesses computational hardness and computational randomness. A key technical insight of our work is that the notion of weight-restricted calibration from the recent literature on algorithm fairness, along with standard computational indistinguishability (known as multiaccuracy in the fairness literature), suffices for proving pseudoentropy characterizations for general entropy notions. This demonstrates the power of weight-restricted calibration to enhance the classic Complexity-Theoretic Regularity Lemma (Trevisan, Tulsiani, and Vadhan, 2009) and Leakage Simulation Lemma (Jetchev and Pietrzak, 2014) and allows us to achieve an exponential improvement in the complexity dependency on the alphabet size compared to the pseudoentropy characterizations by Casacuberta, Dwork, and Vadhan (2024) based on the much stronger notion of multicalibration. We show that the exponential dependency on the alphabet size is inevitable for multicalibration as well as for the weaker notion of calibrated multiaccuracy.

Authors: Lunjia Hu, Salil Vadhan

Pseudoentropy characterizations provide a quantitatively precise demonstration of the close relationship between computational hardness and computational randomness. We prove a unified pseudoentropy characterization that generalizes and strengthens previous results for both uniform and non-uniform models of computation. Our characterization holds for a general family of entropy notions that encompasses the common notions of Shannon entropy and min entropy as special cases. Moreover, we show that the characterizations for different entropy notions can be simultaneously achieved by a single, universal function that simultaneously witnesses computational hardness and computational randomness. A key technical insight of our work is that the notion of weight-restricted calibration from the recent literature on algorithm fairness, along with standard computational indistinguishability (known as multiaccuracy in the fairness literature), suffices for proving pseudoentropy characterizations for general entropy notions. This demonstrates the power of weight-restricted calibration to enhance the classic Complexity-Theoretic Regularity Lemma (Trevisan, Tulsiani, and Vadhan, 2009) and Leakage Simulation Lemma (Jetchev and Pietrzak, 2014) and allows us to achieve an exponential improvement in the complexity dependency on the alphabet size compared to the pseudoentropy characterizations by Casacuberta, Dwork, and Vadhan (2024) based on the much stronger notion of multicalibration. We show that the exponential dependency on the alphabet size is inevitable for multicalibration as well as for the weaker notion of calibrated multiaccuracy.

Wednesday, July 09 2025, 00:00

Complexity Results of Persuasion

Authors: Alban Grastien

We prove that persuasion is an NP-complete problem.

Authors: Alban Grastien

We prove that persuasion is an NP-complete problem.

Wednesday, July 09 2025, 00:00

On the Complexity of Problems on Graphs Defined on Groups

Authors: Bireswar Das, Dipan Dey, Jinia Ghosh

We study the complexity of graph problems on graphs defined on groups, especially power graphs. We observe that an isomorphism invariant problem, such as Hamiltonian Path, Partition into Cliques, Feedback Vertex Set, Subgraph Isomorphism, cannot be NP-complete for power graphs, commuting graphs, enhanced power graphs, directed power graphs, and bounded-degree Cayley graphs, assuming the Exponential Time Hypothesis (ETH). An analogous result holds for isomorphism invariant group problems: no such problem can be NP-complete unless ETH is false. We show that the Weighted Max-Cut problem is NP-complete in power graphs. We also show that, unless ETH is false, the Graph Motif problem cannot be solved in quasipolynomial time on power graphs, even for power graphs of cyclic groups. We study the recognition problem of power graphs when the adjacency matrix or list is given as input and show that for abelian groups and some classes of nilpotent groups, it is solvable in polynomial time.

Authors: Bireswar Das, Dipan Dey, Jinia Ghosh

We study the complexity of graph problems on graphs defined on groups, especially power graphs. We observe that an isomorphism invariant problem, such as Hamiltonian Path, Partition into Cliques, Feedback Vertex Set, Subgraph Isomorphism, cannot be NP-complete for power graphs, commuting graphs, enhanced power graphs, directed power graphs, and bounded-degree Cayley graphs, assuming the Exponential Time Hypothesis (ETH). An analogous result holds for isomorphism invariant group problems: no such problem can be NP-complete unless ETH is false. We show that the Weighted Max-Cut problem is NP-complete in power graphs. We also show that, unless ETH is false, the Graph Motif problem cannot be solved in quasipolynomial time on power graphs, even for power graphs of cyclic groups. We study the recognition problem of power graphs when the adjacency matrix or list is given as input and show that for abelian groups and some classes of nilpotent groups, it is solvable in polynomial time.

Wednesday, July 09 2025, 00:00

Lineal Extensions of Kakeya Sets Missing Every ee-Random Point

Authors: Neil Lutz, Spencer Park Martin, Rain White

By effectivizing a Kakeya set construction, we prove the existence of lines in all directions that do not contain any double exponential time random points. This means each point on these lines has an algorithmically predictable location, to the extent that a gambler in an environment with fair payouts can, using double exponential time computing resources, amass unbounded capital placing bets on increasingly precise estimates of the point's location. Our result resolves an open question published by Lutz and Lutz (2015).

Authors: Neil Lutz, Spencer Park Martin, Rain White

By effectivizing a Kakeya set construction, we prove the existence of lines in all directions that do not contain any double exponential time random points. This means each point on these lines has an algorithmically predictable location, to the extent that a gambler in an environment with fair payouts can, using double exponential time computing resources, amass unbounded capital placing bets on increasingly precise estimates of the point's location. Our result resolves an open question published by Lutz and Lutz (2015).

Wednesday, July 09 2025, 00:00

$k$-means considered harmful: On arbitrary topological changes in Mapper complexes

Authors: Mikael Vejdemo-Johansson

The Mapper construction is one of the most widespread tools from Topological Data Analysis. There is an unfortunate trend as the construction has gained traction to use clustering methods with properties that end up distorting any analysis results from the construction. In this paper we will see a few ways in which widespread choices of clustering algorithms have arbitrarily large distortions of the features visible in the final Mapper complex.

Authors: Mikael Vejdemo-Johansson

The Mapper construction is one of the most widespread tools from Topological Data Analysis. There is an unfortunate trend as the construction has gained traction to use clustering methods with properties that end up distorting any analysis results from the construction. In this paper we will see a few ways in which widespread choices of clustering algorithms have arbitrarily large distortions of the features visible in the final Mapper complex.

Wednesday, July 09 2025, 00:00

Fast and Accurate Collision Probability Estimation for Autonomous Vehicles using Adaptive Sigma-Point Sampling

Authors: Charles Champagne Cossette, Taylor Scott Clawson, Andrew Feit

A novel algorithm is presented for the estimation of collision probabilities between dynamic objects with uncertain trajectories, where the trajectories are given as a sequence of poses with Gaussian distributions. We propose an adaptive sigma-point sampling scheme, which ultimately produces a fast, simple algorithm capable of estimating the collision probability with a median error of 3.5%, and a median runtime of 0.21ms, when measured on an Intel Xeon Gold 6226R Processor. Importantly, the algorithm explicitly accounts for the collision probability's temporal dependence, which is often neglected in prior work and otherwise leads to an overestimation of the collision probability. Finally, the method is tested on a diverse set of relevant real-world scenarios, consisting of 400 6-second snippets of autonomous vehicle logs, where the accuracy and latency is rigorously evaluated.

Authors: Charles Champagne Cossette, Taylor Scott Clawson, Andrew Feit

A novel algorithm is presented for the estimation of collision probabilities between dynamic objects with uncertain trajectories, where the trajectories are given as a sequence of poses with Gaussian distributions. We propose an adaptive sigma-point sampling scheme, which ultimately produces a fast, simple algorithm capable of estimating the collision probability with a median error of 3.5%, and a median runtime of 0.21ms, when measured on an Intel Xeon Gold 6226R Processor. Importantly, the algorithm explicitly accounts for the collision probability's temporal dependence, which is often neglected in prior work and otherwise leads to an overestimation of the collision probability. Finally, the method is tested on a diverse set of relevant real-world scenarios, consisting of 400 6-second snippets of autonomous vehicle logs, where the accuracy and latency is rigorously evaluated.

Wednesday, July 09 2025, 00:00

A Formal Refutation of the Blockchain Trilemma

Authors: Craig Wright

The so-called blockchain trilemma asserts the impossibility of simultaneously achieving scalability, security, and decentralisation within a single blockchain protocol. In this paper, we formally refute that proposition. Employing predicate logic, formal automata theory, computational complexity analysis, and graph-theoretic measures of relay topology--specifically Baran's model of network path redundancy--we demonstrate that the trilemma constitutes a category error, conflates distinct analytical domains, and relies upon unproven causal assumptions. We further expose its reliance on composition fallacies drawn from flawed system implementations. A constructive counterexample is presented: a blockchain protocol exhibiting unbounded transaction throughput, cryptographic security under adversarial load, and multipath decentralised propagation. This example is not hypothetical but grounded in protocol design enabled by compact block relay, SPV verification, and IPv6 multicast. The trilemma is revealed not as a law of protocol architecture, but as a heuristic fallacy sustained by imprecision and design defeatism.

Authors: Craig Wright

The so-called blockchain trilemma asserts the impossibility of simultaneously achieving scalability, security, and decentralisation within a single blockchain protocol. In this paper, we formally refute that proposition. Employing predicate logic, formal automata theory, computational complexity analysis, and graph-theoretic measures of relay topology--specifically Baran's model of network path redundancy--we demonstrate that the trilemma constitutes a category error, conflates distinct analytical domains, and relies upon unproven causal assumptions. We further expose its reliance on composition fallacies drawn from flawed system implementations. A constructive counterexample is presented: a blockchain protocol exhibiting unbounded transaction throughput, cryptographic security under adversarial load, and multipath decentralised propagation. This example is not hypothetical but grounded in protocol design enabled by compact block relay, SPV verification, and IPv6 multicast. The trilemma is revealed not as a law of protocol architecture, but as a heuristic fallacy sustained by imprecision and design defeatism.

Wednesday, July 09 2025, 00:00

Parameterized Restless Temporal Path

Authors: Justine Cauvi, Laurent Viennot

Recently, Bumpus and Meeks introduced a purely temporal parameter, called vertex-interval-membership-width, which is promising for the design of fixed-parameter tractable (FPT) algorithms for vertex reachability problems in temporal graphs. We study this newly introduced parameter for the problem of restless temporal paths, in which the waiting time at each node is restricted. In this article, we prove that, in the interval model, where arcs are present for entire time intervals, finding a restless temporal path is NP-hard even if the vertex-interval-membership-width is equal to three. We exhibit FPT algorithms for the point model, where arcs are present at specific points in time, both with uniform delay one and arbitrary positive delays. In the latter case, this comes with a slight additional computational cost.

Authors: Justine Cauvi, Laurent Viennot

Recently, Bumpus and Meeks introduced a purely temporal parameter, called vertex-interval-membership-width, which is promising for the design of fixed-parameter tractable (FPT) algorithms for vertex reachability problems in temporal graphs. We study this newly introduced parameter for the problem of restless temporal paths, in which the waiting time at each node is restricted. In this article, we prove that, in the interval model, where arcs are present for entire time intervals, finding a restless temporal path is NP-hard even if the vertex-interval-membership-width is equal to three. We exhibit FPT algorithms for the point model, where arcs are present at specific points in time, both with uniform delay one and arbitrary positive delays. In the latter case, this comes with a slight additional computational cost.

Wednesday, July 09 2025, 00:00

An Optimal Algorithm for Shortest Paths in Unweighted Disk Graphs

Authors: Bruce W. Brewer, Haitao Wang

Given in the plane a set $S$ of $n$ points and a set of disks centered at these points, the disk graph $G(S)$ induced by these disks has vertex set $S$ and an edge between two vertices if their disks intersect. Note that the disks may have different radii. We consider the problem of computing shortest paths from a source point $s\in S$ to all vertices in $G(S)$ where the length of a path in $G(S)$ is defined as the number of edges in the path. The previously best algorithm solves the problem in $O(n\log^2 n)$ time. A lower bound of $\Omega(n\log n)$ is also known for this problem under the algebraic decision tree model. In this paper, we present an $O(n\log n)$ time algorithm, which matches the lower bound and thus is optimal. Another virtue of our algorithm is that it is quite simple.

Authors: Bruce W. Brewer, Haitao Wang

Given in the plane a set $S$ of $n$ points and a set of disks centered at these points, the disk graph $G(S)$ induced by these disks has vertex set $S$ and an edge between two vertices if their disks intersect. Note that the disks may have different radii. We consider the problem of computing shortest paths from a source point $s\in S$ to all vertices in $G(S)$ where the length of a path in $G(S)$ is defined as the number of edges in the path. The previously best algorithm solves the problem in $O(n\log^2 n)$ time. A lower bound of $\Omega(n\log n)$ is also known for this problem under the algebraic decision tree model. In this paper, we present an $O(n\log n)$ time algorithm, which matches the lower bound and thus is optimal. Another virtue of our algorithm is that it is quite simple.

Wednesday, July 09 2025, 00:00

Learning-Augmented Online Covering Problems

Authors: Afrouz Jabal Ameli, Laura Sanita, Moritz Venzin

We give a very general and simple framework to incorporate predictions on requests for online covering problems in a rigorous and black-box manner. Our framework turns any online algorithm with competitive ratio $\rho(k, \cdot)$ depending on $k$, the number of arriving requests, into an algorithm with competitive ratio of $\rho(\eta, \cdot)$, where $\eta$ is the prediction error. With accurate enough prediction, the resulting competitive ratio breaks through the corresponding worst-case online lower bounds, and smoothly degrades as the prediction error grows. This framework directly applies to a wide range of well-studied online covering problems such as facility location, Steiner problems, set cover, parking permit, etc., and yields improved and novel bounds.

Authors: Afrouz Jabal Ameli, Laura Sanita, Moritz Venzin

We give a very general and simple framework to incorporate predictions on requests for online covering problems in a rigorous and black-box manner. Our framework turns any online algorithm with competitive ratio $\rho(k, \cdot)$ depending on $k$, the number of arriving requests, into an algorithm with competitive ratio of $\rho(\eta, \cdot)$, where $\eta$ is the prediction error. With accurate enough prediction, the resulting competitive ratio breaks through the corresponding worst-case online lower bounds, and smoothly degrades as the prediction error grows. This framework directly applies to a wide range of well-studied online covering problems such as facility location, Steiner problems, set cover, parking permit, etc., and yields improved and novel bounds.

Wednesday, July 09 2025, 00:00

Instance-Optimal Quantum State Certification with Entangled Measurements

Authors: Ryan O'Donnell, Chirag Wadhwa

We consider the task of quantum state certification: given a description of a hypothesis state $\sigma$ and multiple copies of an unknown state $\rho$, a tester aims to determine whether the two states are equal or $\epsilon$-far in trace distance. It is known that $\Theta(d/\epsilon^2)$ copies of $\rho$ are necessary and sufficient for this task, assuming the tester can make entangled measurements over all copies [CHW07,OW15,BOW19]. However, these bounds are for a worst-case $\sigma$, and it is not known what the optimal copy complexity is for this problem on an instance-by-instance basis. While such instance-optimal bounds have previously been shown for quantum state certification when the tester is limited to measurements unentangled across copies [CLO22,CLHL22], they remained open when testers are unrestricted in the kind of measurements they can perform. We address this open question by proving nearly instance-optimal bounds for quantum state certification when the tester can perform fully entangled measurements. Analogously to the unentangled setting, we show that the optimal copy complexity for certifying $\sigma$ is given by the worst-case complexity times the fidelity between $\sigma$ and the maximally mixed state. We prove our lower bounds using a novel quantum analogue of the Ingster-Suslina method, which is likely to be of independent interest. This method also allows us to recover the $\Omega(d/\epsilon^2)$ lower bound for mixedness testing [OW15], i.e., certification of the maximally mixed state, with a surprisingly simple proof.

Authors: Ryan O'Donnell, Chirag Wadhwa

We consider the task of quantum state certification: given a description of a hypothesis state $\sigma$ and multiple copies of an unknown state $\rho$, a tester aims to determine whether the two states are equal or $\epsilon$-far in trace distance. It is known that $\Theta(d/\epsilon^2)$ copies of $\rho$ are necessary and sufficient for this task, assuming the tester can make entangled measurements over all copies [CHW07,OW15,BOW19]. However, these bounds are for a worst-case $\sigma$, and it is not known what the optimal copy complexity is for this problem on an instance-by-instance basis. While such instance-optimal bounds have previously been shown for quantum state certification when the tester is limited to measurements unentangled across copies [CLO22,CLHL22], they remained open when testers are unrestricted in the kind of measurements they can perform. We address this open question by proving nearly instance-optimal bounds for quantum state certification when the tester can perform fully entangled measurements. Analogously to the unentangled setting, we show that the optimal copy complexity for certifying $\sigma$ is given by the worst-case complexity times the fidelity between $\sigma$ and the maximally mixed state. We prove our lower bounds using a novel quantum analogue of the Ingster-Suslina method, which is likely to be of independent interest. This method also allows us to recover the $\Omega(d/\epsilon^2)$ lower bound for mixedness testing [OW15], i.e., certification of the maximally mixed state, with a surprisingly simple proof.

Wednesday, July 09 2025, 00:00

Non-Adaptive Evaluation of $k$-of-$n$ Functions: Tight Gap and a Unit-Cost PTAS

Authors: Mads Anker Nielsen, Lars Rohwedder, Kevin Schewior

We consider the Stochastic Boolean Function Evaluation (SBFE) problem in the well-studied case of $k$-of-$n$ functions: There are independent Boolean random variables $x_1,\dots,x_n$ where each variable $i$ has a known probability $p_i$ of taking value $1$, and a known cost $c_i$ that can be paid to find out its value. The value of the function is $1$ iff there are at least $k$ $1$s among the variables. The goal is to efficiently compute a strategy that, at minimum expected cost, tests the variables until the function value is determined. While an elegant polynomial-time exact algorithm is known when tests can be made adaptively, we focus on the non-adaptive variant, for which much less is known. First, we show a clean and tight lower bound of $2$ on the adaptivity gap, i.e., the worst-case multiplicative loss in the objective function caused by disallowing adaptivity, of the problem. This improves the tight lower bound of $3/2$ for the unit-cost variant. Second, we give a PTAS for computing the best non-adaptive strategy in the unit-cost case, the first PTAS for an SBFE problem. At the core, our scheme establishes a novel notion of two-sided dominance (w.r.t. the optimal solution) by guessing so-called milestone tests for a set of carefully chosen buckets of tests. To turn this technique into a polynomial-time algorithm, we use a decomposition approach paired with a random-shift argument.

Authors: Mads Anker Nielsen, Lars Rohwedder, Kevin Schewior

We consider the Stochastic Boolean Function Evaluation (SBFE) problem in the well-studied case of $k$-of-$n$ functions: There are independent Boolean random variables $x_1,\dots,x_n$ where each variable $i$ has a known probability $p_i$ of taking value $1$, and a known cost $c_i$ that can be paid to find out its value. The value of the function is $1$ iff there are at least $k$ $1$s among the variables. The goal is to efficiently compute a strategy that, at minimum expected cost, tests the variables until the function value is determined. While an elegant polynomial-time exact algorithm is known when tests can be made adaptively, we focus on the non-adaptive variant, for which much less is known. First, we show a clean and tight lower bound of $2$ on the adaptivity gap, i.e., the worst-case multiplicative loss in the objective function caused by disallowing adaptivity, of the problem. This improves the tight lower bound of $3/2$ for the unit-cost variant. Second, we give a PTAS for computing the best non-adaptive strategy in the unit-cost case, the first PTAS for an SBFE problem. At the core, our scheme establishes a novel notion of two-sided dominance (w.r.t. the optimal solution) by guessing so-called milestone tests for a set of carefully chosen buckets of tests. To turn this technique into a polynomial-time algorithm, we use a decomposition approach paired with a random-shift argument.

Wednesday, July 09 2025, 00:00

25 Additional Problems -- Extension to the Book "125 Problems in Text Algorithms"

Authors: Maxime Crochemore, Thierry Lecroq, Wojtek Rytter

This very preliminary text is related to ``Algorithms on Texts'', also called ``Algorithmic Stringology''. It is an extension of the book ``125 Problems in Text Algorithms'' providing, in the same compact style, more problems with solutions. We refer also to the companions to ``Text algorithms'' available at monge.univ-mlv.fr/~mac/CLR/clr1-20.pdf and at the web page 125-problems.univ-mlv.fr, where all 150 problems (including the ones presented here) are briefly announced. The selected problems satisfy three criteria: challenging, having short tricky solutions and solvable with only very basic background in stringology. For the basics in stringology we refer to monge.univ-mlv.fr/~mac/CLR/clr1-20.pdf.

Authors: Maxime Crochemore, Thierry Lecroq, Wojtek Rytter

This very preliminary text is related to ``Algorithms on Texts'', also called ``Algorithmic Stringology''. It is an extension of the book ``125 Problems in Text Algorithms'' providing, in the same compact style, more problems with solutions. We refer also to the companions to ``Text algorithms'' available at http://monge.univ-mlv.fr/~mac/CLR/clr1-20.pdf and at the web page http://125-problems.univ-mlv.fr, where all 150 problems (including the ones presented here) are briefly announced. The selected problems satisfy three criteria: challenging, having short tricky solutions and solvable with only very basic background in stringology. For the basics in stringology we refer to http://monge.univ-mlv.fr/~mac/CLR/clr1-20.pdf.

Wednesday, July 09 2025, 00:00

What is Intelligence? Layers of Emergence

from Nisheeth Vishnoi

How Intelligence Arises from Nature, One Layer at a Time The Trouble with Definitions “The Tao that can be named is not the eternal Tao.” — Lao Tzu As a mathematician, I’ve long sought clean definitions. Much of my work involves building precise frameworks — starting by defining key concepts, isolating the core of a […]

How Intelligence Arises from Nature, One Layer at a Time

The Trouble with Definitions

“The Tao that can be named is not the eternal Tao.” — Lao Tzu

As a mathematician, I’ve long sought clean definitions. Much of my work involves building precise frameworks — starting by defining key concepts, isolating the core of a problem, formalizing it, and tracing its implications to their logical end.

Yet over time, I’ve come to see not just the limits of definitions, but their quiet distortions—the way they can flatten nuance in the name of clarity. The richness of a living idea gets traded for the sterile comfort of formal neatness. Sometimes, defining isn’t just clarifying — it’s an act of power: shaping perception, and granting authority to the one who defines.

Few ideas reveal this tension more vividly than intelligence. We talk about it as if we know what it is — a score, a skill, a spark. But what is it, really? And can something so dynamic ever be pinned down?

I think of intelligence not as a fixed trait, but as an experience — not unlike beauty — arising in context, felt through interaction.

So while we try to define intelligence — because we must — to witness it, to live with it, or to build systems that move with it, we need something else: humility. An attention to context. A willingness to recognize that intelligence, like beauty, is often messy, partial, plural, heuristic, and still astonishingly effective.

But even our capacity to see intelligence is shaped by history. In The Myth of Superintelligence, I argued that our attempts to define intelligence are never neutral. They reflect what we choose to measure, optimize, and reward. This essay is not a repetition of that critique. It is a step back. A shift in lens. It asks not what intelligence is, but when and how it arises—not as a trait, but as something unfolding across time, scale, and structure.

Because the power to define has always been the power to exclude. Colonial systems didn’t just extract labor and land—they imposed ways of seeing. In doing so, they dismissed the intelligence embedded in other ways of knowing, reframing rich knowledge traditions as myth or superstition. These distortions still echo in how we define and measure intelligence today. African polyrhythms were labeled primitive. Classical Indian music was exoticized or ignored. Indigenous knowledge systems—deeply attuned to land, season, and cycle—were reduced to folklore. Intelligence was there. But the lens refused to see it.

This is why any inquiry into intelligence must also be an inquiry into perspective. Definitions don’t just clarify. They constrain. They shape not only what we see, but what we believe intelligence can be.

This series is an attempt to widen the lens—to trace intelligence not as a fixed trait, but as a dynamic unfolding across layers of complexity. We begin with the silent elegance of physical systems, where matter flows under law, solving problems through coherence and constraint. From there, we enter the domain of evolution, where life adapts through variation and feedback, accumulating structure over time. We then move to the responsive intelligence of behavior—organisms without minds that nonetheless solve, coordinate, and learn through interaction.

But these are just the foundations. In the second half, we abstract upward: tracing how intelligence evolves the ability to frame problems, to reflect on and revise its own rules, and finally, to orient itself—to choose what matters. This is where intelligence becomes recursive, contextual, and ultimately, meaningful. Not just a solver of problems, but a seeker of value.

Read the full essay by subscribing (for free) to The Intelligence Loop.

By nisheethvishnoi

Tuesday, July 08 2025, 11:35

Proof Complexity 2025

from CS Theory Events

August 11-13, 2025 Oxford, UK feasible-math.org/events/PC25/ Proof complexity is a vibrant area in the intersection of computational complexity, algorithms and mathematical logic exploring the inherent difficulty of proving statements in different formal proof systems. This workshop aims to cover both traditional topics and emerging trends in the field, including lower bounds on lengths of proofs, … Continue reading Proof Complexity 2025

By shacharlovett

August 11-13, 2025 Oxford, UK https://feasible-math.org/events/PC25/ Proof complexity is a vibrant area in the intersection of computational complexity, algorithms and mathematical logic exploring the inherent difficulty of proving statements in different formal proof systems. This workshop aims to cover both traditional topics and emerging trends in the field, including lower bounds on lengths of proofs, … Continue reading Proof Complexity 2025

By shacharlovett

Tuesday, July 08 2025, 00:48

Testing for Renamability to Classes of Clause Sets

Authors: Albert Brandl, Christian G. Fermüller, Gernot Salzer

This paper investigates the problem of testing clause sets for membership in classes known from literature. In particular, we are interested in classes defined via renaming: Is it possible to rename the predicates in a way such that positive and negative literals satisfy certain conditions? We show that for classes like Horn or OCC1N, the existence of such renamings can be decided in polynomial time, whereas the same problem is NP-complete for class PVD. The decision procedures are based on hyper-resolution; if a renaming exists, it can be extracted from the final saturated clause set.

Authors: Albert Brandl, Christian G. Fermüller, Gernot Salzer

This paper investigates the problem of testing clause sets for membership in classes known from literature. In particular, we are interested in classes defined via renaming: Is it possible to rename the predicates in a way such that positive and negative literals satisfy certain conditions? We show that for classes like Horn or OCC1N, the existence of such renamings can be decided in polynomial time, whereas the same problem is NP-complete for class PVD. The decision procedures are based on hyper-resolution; if a renaming exists, it can be extracted from the final saturated clause set.

Tuesday, July 08 2025, 00:00

Low sets for counting functions

Authors: Yaroslav Ivanashev

In this paper, we characterize the classes of languages and functions that are low for counting function classes. The classes #P and GapP have their low classes exactly characterized: Low(#P) = UP $\cap$ coUP and Low(GapP) = SPP. We prove that Low(TotP) = P, Low(SpanP) = NP $\cap$ coNP, and give characterizations of low function classes for #P, GapP, TotP, and SpanP. We establish the relations between NPSVt, UPSVt, and the counting function classes. For each of the inclusions between these classes we give an equivalent inclusion between language classes. We also prove that SpanP $\subseteq$ GapP if and only if NP $\subseteq$ SPP, and the inclusion GapP+ $\subseteq$ SpanP implies PH = $\Sigma_{2}^{P}$. For the classes #P, GapP, TotP, and SpanP we summarize the known results and prove that each of these classes is closed under left composition with FP+ if and only if it collapses to its low class of functions.

Authors: Yaroslav Ivanashev

In this paper, we characterize the classes of languages and functions that are low for counting function classes. The classes #P and GapP have their low classes exactly characterized: Low(#P) = UP $\cap$ coUP and Low(GapP) = SPP. We prove that Low(TotP) = P, Low(SpanP) = NP $\cap$ coNP, and give characterizations of low function classes for #P, GapP, TotP, and SpanP. We establish the relations between NPSVt, UPSVt, and the counting function classes. For each of the inclusions between these classes we give an equivalent inclusion between language classes. We also prove that SpanP $\subseteq$ GapP if and only if NP $\subseteq$ SPP, and the inclusion GapP+ $\subseteq$ SpanP implies PH = $\Sigma_{2}^{P}$. For the classes #P, GapP, TotP, and SpanP we summarize the known results and prove that each of these classes is closed under left composition with FP+ if and only if it collapses to its low class of functions.

Tuesday, July 08 2025, 00:00

PFCS: Prime Factorization Cache System for Deterministic Data Relationship Discovery

Authors: Duy Le

Cache systems fundamentally limit modern computing performance due to their inability to precisely capture data relationships. While achieving 85-92% hit rates, traditional systems rely on statistical heuristics that cannot guarantee relationship discovery, leading to suboptimal prefetching and resource waste. We present PFCS (Prime Factorization Cache System), which leverages the mathematical uniqueness of prime factorization to achieve deterministic relationship discovery with zero false positives. PFCS assigns unique primes to data elements and represents relationships as composite numbers, enabling the recovery of perfect relationships through factorization. A comprehensive evaluation across database, ML, and HPC workloads demonstrates an average performance improvement of x 6.2, 98.9% hit rates, and a 38% power reduction compared to state-of-the-art systems. The mathematical foundation provides formal guarantees impossible with approximation-based approaches, establishing a new paradigm for cache system design

Authors: Duy Le

Cache systems fundamentally limit modern computing performance due to their inability to precisely capture data relationships. While achieving 85-92% hit rates, traditional systems rely on statistical heuristics that cannot guarantee relationship discovery, leading to suboptimal prefetching and resource waste. We present PFCS (Prime Factorization Cache System), which leverages the mathematical uniqueness of prime factorization to achieve deterministic relationship discovery with zero false positives. PFCS assigns unique primes to data elements and represents relationships as composite numbers, enabling the recovery of perfect relationships through factorization. A comprehensive evaluation across database, ML, and HPC workloads demonstrates an average performance improvement of x 6.2, 98.9% hit rates, and a 38% power reduction compared to state-of-the-art systems. The mathematical foundation provides formal guarantees impossible with approximation-based approaches, establishing a new paradigm for cache system design

Tuesday, July 08 2025, 00:00

Approximation and Hardness of Polychromatic TSP

Authors: Thomas Schibler, Subhash Suri, Jie Xue

We introduce the Polychromatic Traveling Salesman Problem (PCTSP), where the input is an edge weighted graph whose vertices are partitioned into $k$ equal-sized color classes, and the goal is to find a minimum-length Hamiltonian cycle that visits the classes in a fixed cyclic order. This generalizes the Bipartite TSP (when $k = 2$) and the classical TSP (when $k = n$). We give a polynomial-time $(3 - 2 * 10^{-36})$-approximation algorithm for metric PCTSP. Complementing this, we show that Euclidean PCTSP is APX-hard even in $R^2$, ruling out the existence of a PTAS unless P = NP.

Authors: Thomas Schibler, Subhash Suri, Jie Xue

We introduce the Polychromatic Traveling Salesman Problem (PCTSP), where the input is an edge weighted graph whose vertices are partitioned into $k$ equal-sized color classes, and the goal is to find a minimum-length Hamiltonian cycle that visits the classes in a fixed cyclic order. This generalizes the Bipartite TSP (when $k = 2$) and the classical TSP (when $k = n$). We give a polynomial-time $(3 - 2 * 10^{-36})$-approximation algorithm for metric PCTSP. Complementing this, we show that Euclidean PCTSP is APX-hard even in $R^2$, ruling out the existence of a PTAS unless P = NP.

Tuesday, July 08 2025, 00:00

Node-neighbor subnetworks and Hk-core decomposition