Theory of Computing Report

Traditionally, protocol performance is summarized using two metrics: latency (measured in rounds), and communication complexity (measured asymptotically, e.g., $O(n^2)$). If both are small, we might expect the protocol to perform well in practice. But this intuition is incomplete. When Low Latency and Low Communication Aren’t Enough Consider two protocols: Protocol...

Authors: Cornelius Brand, Radu Curticapean, Petteri Kaski, Baitian Li, Ian Orzel, Tim Seppelt, Jiaheng Wang

The complexity of bilinear maps (equivalently, of $3$-mode tensors) has been studied extensively, most notably in the context of matrix multiplication. While circuit complexity and tensor rank coincide asymptotically for $3$-mode tensors, this correspondence breaks down for $d \geq 4$ modes. As a result, the complexity of $d$-mode tensors for larger fixed $d$ remains poorly understood, despite its relevance, e.g., in fine-grained complexity. Our paper explores this intermediate regime. First, we give a "graph-theoretic" proof of Strassen's $2ω/3$ bound on the asymptotic rank exponent of $3$-mode tensors. Our proof directly generalizes to an upper bound of $(d-1)ω/3$ for $d$-mode tensors. Using refined techniques available only for $d\geq 4$ modes, we improve this bound beyond the current state of the art for $ω$. We also obtain a bound of $d/2+1$ on the asymptotic exponent of circuit complexity of generic $d$-mode tensors and optimized bounds for $d \in \{4,5\}$. To the best of our knowledge, asymptotic circuit complexity (rather than rank) of tensors has not been studied before. To obtain a robust theory, we first ask whether low complexity of $T$ and $U$ imply low complexity of their Kronecker product $T \otimes U$. While this crucially holds for rank (and thus for circuit complexity in $3$ modes), we show that assumptions from fine-grained complexity rule out such a submultiplicativity for the circuit complexity of tensors with many modes. In particular, assuming the Hyperclique Conjecture, this failure occurs already for $d=8$ modes. Nevertheless, we can salvage a restricted notion of submultiplicativity. From a technical perspective, our proofs heavily make use of the graph tensors $T_H$, as employed by Christandl and Zuiddam ({\em Comput.~Complexity}~28~(2019)~27--56) and [...]

Authors: Grzegorz Gutowski, Mikołaj Rams

For a directed graph $G$, and a linear order $\ll$ on the vertices of $G$, we define backedge graph $G^\ll$ to be the undirected graph on the same vertex set with edge $\{u,w\}$ in $G^\ll$ if and only if $(u,w)$ is an arc in $G$ and $w \ll u$. The directed clique number of a directed graph $G$ is defined as the minimum size of the maximum clique in the backedge graph $G^\ll$ taken over all linear orders $\ll$ on the vertices of $G$. A natural computational problem is to decide for a given directed graph $G$ and a positive integer $t$, if the directed clique number of $G$ is at most $t$. This problem has polynomial algorithm for $t=1$ and is known to be \NP-complete for every fixed $t\ge3$, even for tournaments. In this note we prove that this problem is $Σ^\mathsf{P}_{2}$-complete when $t$ is given on the input.

Authors: Simon Gmeiner, Andreas S. Schulz

How can a stack of identical blocks be arranged to extend beyond the edge of a table as far as possible? We consider a generalization of this classic puzzle to blocks that differ in width and mass. Despite the seemingly simple premise, we demonstrate that it is unlikely that one can efficiently determine a stack configuration of maximum overhang. Formally, we prove that the Block-Stacking Problem is NP-hard, partially answering an open question from the literature. Furthermore, we demonstrate that the restriction to stacks without counterweights has a surprising connection to the Airplane Refueling Problem, another famous puzzle, and to Robust Appointment Scheduling, a problem of practical relevance. In addition to revealing a remarkable relation to the real-world challenge of devising schedules under uncertainty, their equivalence unveils a polynomial-time approximation scheme, that is, a $(1+ε)$-approximation algorithm, for Block Stacking without counterbalancing and a $(2+ε)$-approximation algorithm for the general case.

Authors: Austen Fan, Jin-Yi Cai, Shuai Shao, Zhuxiao Tang

We prove a complete complexity classification theorem for the planar eight-vertex model. For every parameter setting in ${\mathbb C}$ for the eight-vertex model, the partition function is either (1) computable in P-time for every graph, or (2) \#P-hard for general graphs but computable in P-time for planar graphs, or (3) \#P-hard even for planar graphs. The classification has an explicit criterion. In (2), we discover new P-time computable eight-vertex models on planar graphs beyond Kasteleyn's algorithm for counting planar perfect matchings. They are obtained by a combinatorial transformation to the planar {\sc Even Coloring} problem followed by a holographic transformation to the tractable cases in the planar six-vertex model. In the process, we also encounter non-local connections between the planar eight vertex model and the bipartite Ising model, conformal lattice interpolation and Möbius transformation from complex analysis. The proof also makes use of cyclotomic fields.

Authors: Giannis Spiliopoulos, Alexandros Troupiotis-Kapeliaris, Kostas Patroumpas, Nikolaos Liapis, Dimitrios Skoutas, Dimitris Zissis, Nikos Bikakis

Modeling vessel activity at sea is critical for a wide range of applications, including route planning, transportation logistics, maritime safety, and environmental monitoring. Over the past two decades, the Automatic Identification System (AIS) has enabled real-time monitoring of hundreds of thousands of vessels, generating huge amounts of data daily. One major challenge in using AIS data is the presence of large gaps in vessel trajectories, often caused by coverage limitations or intentional transmission interruptions. These gaps can significantly degrade data quality, resulting in inaccurate or incomplete analysis. State-of-the-art imputation approaches have mainly been devised to tackle gaps in vehicle trajectories, even when the underlying road network is not considered. But the motion patterns of sailing vessels differ substantially, e.g., smooth turns, maneuvering near ports, or navigating in adverse weather conditions. In this application paper, we propose HABIT, a lightweight, configurable H3 Aggregation-Based Imputation framework for vessel Trajectories. This data-driven framework provides a valuable means to impute missing trajectory segments by extracting, analyzing, and indexing motion patterns from historical AIS data. Our empirical study over AIS data across various timeframes, densities, and vessel types reveals that HABIT produces maritime trajectory imputations performing comparably to baseline methods in terms of accuracy, while performing better in terms of latency while accounting for vessel characteristics and their motion patterns.

Authors: Alessandro Epasto, Xin Lyu, Pasin Manurangsi

We study the computational cost of differential privacy in terms of memory efficiency. While the trade-off between accuracy and differential privacy is well-understood, the inherent cost of privacy regarding memory use remains largely unexplored. This paper establishes for the first time an unconditional space lower bound for user-level differential privacy by introducing a novel proof technique based on a multi-player communication game. Central to our approach, this game formally links the hardness of low-memory private algorithms to the necessity of ``contribution capping'' -- tracking and limiting the users who disproportionately impact the dataset. We demonstrate that winning this communication game requires transmitting information proportional to the number of over-active users, which translates directly to memory lower bounds. We apply this framework, as an example, to the fundamental problem of estimating the number of distinct elements in a stream and we prove that any private algorithm requires almost $\widetildeΩ(T^{1/3})$ space to achieve certain error rates in a promise variant of the problem. This resolves an open problem in the literature (by Jain et al. NeurIPS 2023 and Cummings et al. ICML 2025) and establishes the first exponential separation between the space complexity of private algorithms and their non-private $\widetilde{O}(1)$ counterparts for a natural statistical estimation task. Furthermore, we show that this communication-theoretic technique generalizes to broad classes of problems, yielding lower bounds for private medians, quantiles, and max-select.

Authors: Antoine Amarilli, Claire David, Nadime Francis, Victor Marsault, Mikaël Monet, Yann Strozecki

We give the first two algorithms to enumerate all binary words of $\{0,1\}^\ell$ (like Gray codes) while ensuring that the delay and the auxiliary space is independent from $\ell$, i.e., constant time for each word, and constant memory in addition to the $\ell$ bits storing the current word. Our algorithms are given in two new computational models: tape machines and deque machines. We also study more restricted models, queue machines and stack machines, and show that they cannot enumerate all binary words with constant auxiliary space, even with unrestricted delay. A tape machine is a Turing machine that stores the current binary word on a single working tape of length $\ell$. The machine has a single head and must edit its tape to reach all possible words of $\{0,1\}^{\ell}$ , and output them (in unit time, by entering special output states), with no duplicates. We construct a tape machine that achieves this task with constant delay between consecutive outputs, which implies that the machine implements a so-called skew-tolerant quasi-Gray code. We then construct a more involved tape machine that implements a Gray code. A deque machine stores the current binary word on a double-ended queue of length $\ell$, and stores a constant-size internal state. It works as a tape machine, except that it modifies the content of the deque by performing push and pop operations on the endpoints. We construct deque machines that enumerate all words of $\{0,1\}^\ell$ with constant-delay. The main technical challenge in this model is to correctly detect when enumeration has finished. Our work on deque machine is also motivated by other contexts in which endpoint modifications occur naturally. In particular, our result is a first step towards enumerating walks in directed graphs with constant delay and constant auxiliary space, addressing a core task in modern graph database query processing.

Authors: Althaf P, Amit Chattopadhyay, Osamu Saeki

A merge tree is a fundamental topological structure used to capture the sub-level set (and similarly, super-level set) topology in scalar data analysis. The interleaving distance is a theoretically sound, stable metric for comparing merge trees. However, computing this distance exactly is NP-hard. First fixed-parameter tractable (FPT) algorithm for it's exact computation introduces the concept of an $\varepsilon$-good map between two merge trees, where $\varepsilon$ is a candidate value for the interleaving distance. The complexity of their algorithm is $O(2^{2τ}(2τ)^{2τ+2}\cdot n^2\log^3n)$ where $τ$ is the degree-bound parameter and $n$ is the total number of nodes in both the merge trees. Their algorithm exhibits exponential complexity in $τ$, which increases with the increasing value of $\varepsilon$. In the current paper, we propose an improved FPT algorithm for computing the $\varepsilon$-good map between two merge trees. Our algorithm introduces two new parameters, $η_f$ and $η_g$, corresponding to the numbers of leaf nodes in the merge trees $M_f$ and $M_g$, respectively. This parametrization is motivated by the observation that a merge tree can be decomposed into a collection of unique leaf-to-root paths. The proposed algorithm achieves a complexity of $O\!\left(n^2\log n+η_g^{η_f}(η_f+η_g)\, n \log n \right)$. To obtain this reduced complexity, we assume that number of possible $\varepsilon$-good maps from $M_f$ to $M_g$ does not exceed that from $M_g$ to $M_f$. Notably, the parameters $η_f$ and $η_g$ are independent of the choice of $\varepsilon$. Compared to their algorithm, our approach substantially reduces the search space for computing an optimal $\varepsilon$-good map. We also provide a formal proof of correctness for the proposed algorithm.

Authors: David Shmoys, Varun Suriyanarayana, Seeun William Umboh

The Joint Replenishment Problem (JRP) is a classical inventory management problem, that aims to model the trade-off between coordinating orders for multiple commodities (and their cost) with holding costs incurred by meeting demand in advance. Moseley, Niaparast and Ravi introduced a natural online generalization of the JRP in which inventory corresponding to demands may be replenished late, for a delay cost, or early, for a holding cost. They established that when the holding and delay costs are monotone and uniform across demands, there is a 30-competitive algorithm that employs a greedy strategy and a dual-fitting based analysis. We develop a 5-competitive algorithm that handles arbitrary monotone demand-specific holding and delay cost functions, thus simultaneously improving upon the competitive ratio and relaxing the uniformity assumption. Our primal-dual algorithm is in the spirit of the work Buchbinder, Kimbrel, Levi, Makarychev, and Sviridenko, which maintains a wavefront dual solution to decide when to place an order and which items to order. The main twist is in deciding which requests to serve early. In contrast to the work of Moseley et al., which ranks early requests in ascending order of desired service time and serves them until their total holding cost matches the ordering cost incurred for that item, we extend to the non-uniform case by instead ranking in ascending order of when the delay cost of a demand would reach its current holding cost. An important special case of the JRP is the single-item lot-sizing problem. Here, Moseley et al. gave a 3-competitive algorithm when the holding and delay costs are uniform across demands. We provide a new algorithm for which the competitive ratio is $φ+1 \approx 2.681$, where $φ$ is the golden ratio, which again holds for arbitrary monotone holding-delay costs.

Authors: Daniele Carnevale, Gianlorenzo D'Angelo

Temporal graphs represent networks in which connections change over time, with edges available only at specific moments. Motivated by applications in logistics, multi-agent information spreading, and wireless networks, we introduce the D-Temporal Multi-Broadcast (D-TMB) problem, which asks for scheduling the availability of edges so that a predetermined subset of sources reach all other vertices while optimizing the worst-case temporal distance D from any source. We show that D-TMB generalizes ReachFast (arXiv:2112.08797). We then characterize the computational complexity and approximability of D-TMB under six definitions of temporal distance D, namely Earliest-Arrival (EA), Latest-Departure (LD), Fastest-Time (FT), Shortest-Traveling (ST), Minimum-Hop (MH), and Minimum-Waiting (MW). For a single source, we show that D-TMB can be solved in polynomial time for EA and LD, while for the other temporal distances it is NP-hard and hard to approximate within a factor that depends on the adopted distance function. We give approximation algorithms for FT and MW. For multiple sources, if feasibility is not assumed a priori, the problem is inapproximable within any factor unless P = NP, even with just two sources. We complement this negative result by identifying structural conditions that guarantee tractability for EA and LD for any number of sources.

Authors: Michael A. Bender, William Kuszmaul, Elaine Shi, Rose Silver

We give a (strongly) history-independent two-choice balls-and-bins algorithm on $n$ bins that supports both insertions and deletions on a set of up to $m$ balls, while guaranteeing a maximum load of $m / n + O(1)$ with high probability, and achieving an expected recourse of $O(\log \log (m/n))$ per operation. To the best of our knowledge, this is the first history-independent solution to achieve nontrivial guarantees of any sort for $m/n \ge ω(1)$ and is the first fully dynamic solution (history independent or not) to achieve $O(1)$ overload with $o(m/n)$ expected recourse.

Authors: Mirabel Mendoza-Cadena, Arturo Merino, Mads Anker Nielsen, Kevin Schewior

This paper introduces a framework for combinatorial variants of perpetual-scheduling problems. Given a set system $(E,\mathcal{I})$, a schedule consists of an independent set $I_t \in \mathcal{I}$ for every time step $t \in \mathbb{N}$, with the objective of fulfilling frequency requirements on the occurrence of elements in $E$. We focus specifically on combinatorial bamboo garden trimming, where elements accumulate height at growth rates $g(e)$ for $e \in E$ given as a convex combination of incidence vectors of $\mathcal{I}$ and are reset to zero when scheduled, with the goal of minimizing the maximum height attained by any element. Using the integrality of the matroid-intersection polytope, we prove that, when $(E,\mathcal{I})$ is a matroid, it is possible to guarantee a maximum height of at most 2, which is optimal. We complement this existential result with efficient algorithms for specific matroid classes, achieving a maximum height of 2 for uniform and partition matroids, and 4 for graphic and laminar matroids. In contrast, we show that for general set systems, the optimal guaranteed height is $Θ(\log |E|)$ and can be achieved by an efficient algorithm. For combinatorial pinwheel scheduling, where each element $e\in E$ needs to occur in the schedule at least every $a_e \in \mathbb{N}$ time steps, our results imply bounds on the density sufficient for schedulability.

Authors: R. Jay Martin

We formalize a constructive subclass of locality-preserving deterministic operators acting on graph-indexed state systems. We define the class of Bounded Local Generator Classes (BLGC), consisting of finite-range generators operating on bounded state spaces under deterministic composition. Within this class, incremental update cost is independent of total system dimension. We prove that, under the BLGC assumptions, per-step operator work satisfies W_t = O(1) as the number of nodes M \to \infty, establishing a structural decoupling between global state size and incremental computational effort. The framework admits a Hilbert-space embedding in \ell^2(V; \mathbb{R}^d) and yields bounded operator norms on admissible subspaces. The result applies specifically to the defined subclass and does not claim universality beyond the stated locality and boundedness constraints.

Authors: Ta Duy Nguyem, Alina Ene, Huy Le Nguyen

We study $(ε,δ)$-differentially private algorithms for the problem of approximately computing the top singular vector of a matrix $A\in\mathbb{R}^{n\times d}$ where each row of $A$ is a datapoint in $\mathbb{R}^{d}$. In our privacy model, neighboring inputs differ by one single row/datapoint. We study the private variant of the power iteration method, which is widely adopted in practice. Our algorithm is based on a filtering technique which adapts to the coherence parameter of the input matrix. This technique provides a utility that goes beyond the worst-case guarantees for matrices with low coherence parameter. Our work departs from and complements the work by Hardt-Roth (STOC 2013) which designed a private power iteration method for the privacy model where neighboring inputs differ in one single entry by at most 1.

Authors: Ioannis Caragiannis, Vasilis Gkatzelis, Sebastian Homrighausen

In a setting where $m$ items need to be partitioned among $n$ agents, we evaluate the performance of mechanisms that take as input each agent's \emph{ordinal preferences}, i.e., their ranking of the items from most- to least-preferred. The standard measure for evaluating ordinal mechanisms is the \emph{distortion}, and the vast majority of the literature on distortion has focused on worst-case analysis, leading to some overly pessimistic results. We instead evaluate the distortion of mechanisms with respect to their expected performance when the agents' preferences are generated stochastically. We first show that no ordinal mechanism can achieve a distortion better than $e/(e-1)\approx 1.582$, even if each agent needs to receive exactly one item (i.e., $m=n$) and every agent's values for different items are drawn i.i.d.\ from the same known distribution. We then complement this negative result by proposing an ordinal mechanism that achieves the optimal distortion of $e/(e-1)$ even if each agent's values are drawn from an agent-specific distribution that is unknown to the mechanism. To further refine our analysis, we also optimize the \emph{distortion gap}, i.e., the extent to which an ordinal mechanism approximates the optimal distortion possible for the instance at hand, and we propose a mechanism with a near-optimal distortion gap of $1.076$. Finally, we also evaluate the distortion and distortion gap of simple mechanisms that have a one-pass structure.

Authors: M. Szusterman

This paper investigates the extension complexity of polytopes by exploiting the correspondence between non-negative factorizations of slack matrices and randomized communication protocols. We introduce a geometric characterization of extension complexity based on the width of Markovian protocols, as a variant of the framework introduced by Faenza et al. This enables us to derive a new upper bound of $\tilde{O}(n^3\cdot 1.5^n)$ for the extension complexity of the matching polytope $P_{\text{match}}(n)$, improving upon the standard $2^n$-bound given by Edmonds' description. Additionally, we recover Goemans' compact formulation for the permutahedron using a one-round protocol based on sorting networks.

Authors: Yael Kirkpatrick, John Kuszmaul, Surya Mathialagan, Virginia Vassilevska Williams

We consider the classic 3SUM problem: given sets of integers $A, B, C $, determine whether there is a tuple $(a, b, c) \in A \times B \times C$ satisfying $a + b + c = 0$. The 3SUM Hypothesis, central in fine-grained complexity, states that there does not exist a truly subquadratic time 3SUM algorithm. Given this long-standing barrier, recent work over the past decade has explored 3SUM from a data structural perspective. Specifically, in the 3SUM in preprocessed universes regime, we are tasked with preprocessing sets $A, B$ of size $n$, to create a space-efficient data structure that can quickly answer queries, each of which is a 3SUM problem of the form $A', B', C'$, where $A' \subseteq A$ and $B' \subseteq B$. A series of results have achieved $\tilde{O}(n^2)$ preprocessing time, $\tilde{O}(n^2)$ space, and query time improving progressively from $\tilde{O}(n^{1.9})$ [CL15] to $\tilde{O}(n^{11/6})$ [CVX23] to $\tilde{O}(n^{1.5})$ [KPS25]. Given these series of works improving query time, a natural open question has emerged: can one achieve both truly subquadratic space and truly subquadratic query time for 3SUM in preprocessed universes? We resolve this question affirmatively, presenting a tradeoff curve between query and space complexity. Specifically, we present a simple randomized algorithm achieving $\tilde{O}(n^{1.5 + \varepsilon})$ query time and $\tilde{O}(n^{2 - 2\varepsilon/3})$ space complexity. Furthermore, our algorithm has $\tilde{O}(n^2)$ preprocessing time, matching past work. Notably, quadratic preprocessing is likely necessary for our tradeoff as either the preprocessing or the query time must be at least $n^{2-o(1)}$ under the 3SUM Hypothesis.

Authors: Sushmita Gupta, Pallavi Jain, Sanjay Seetharaman, Meirav Zehavi

We consider the fair division of indivisible items among $n$ agents with additive non-negative normalized valuations, with the goal of obtaining high value guarantees, that is, close to the proportional share for each agent. We prove that partitions where \emph{every} part yields high value for each agent are asymptotically limited by a discrepancy barrier of $Θ(\sqrt{n})$. Guided by this, our main objective is to overcome this barrier and achieve stronger individual guarantees for each agent in polynomial time. Towards this, we are able to exhibit an exponential improvement over the discrepancy barrier. In particular, we can create partitions on-the-go such that when agents arrive sequentially (representing a previously-agreed priority order) and pick a part autonomously and rationally (i.e., one of highest value), then each is guaranteed a part of value at least $\mathsf{PROP} - \mathcal{O}{(\log n)}$. Moreover, we show even better guarantees for three restricted valuation classes such as those defined by: a common ordering on items, a bound on the multiplicity of values, and a hypergraph with a bound on the \emph{influence} of any agent. Specifically, we study instances where: (1) the agents are ``close'' to unanimity in their relative valuation of the items -- a generalization of the ordered additive setting; (2) the valuation functions do not assign the same positive value to more than $t$ items; and (3) the valuation functions respect a hypergraph, a setting introduced by Christodoulou et al. [EC'23], where agents are vertices and items are hyperedges. While the sizes of the hyperedges and neighborhoods can be arbitrary, the influence of any agent $a$, defined as the number of its neighbors who value at least one item positively that $a$ also values positively, is bounded.

Authors: Jonas Ellert, Tomasz Kociumaka

A key principle in string processing is local consistency: using short contexts to handle matching fragments of a string consistently. String synchronizing sets [Kempa, Kociumaka; STOC 2019] are an influential instantiation of this principle. A $τ$-synchronizing set of a length-$n$ string is a set of $O(n/τ)$ positions, chosen via their length-$2τ$ contexts, such that (outside highly periodic regions) at least one position in every length-$τ$ window is selected. Among their applications are faster algorithms for data compression, text indexing, and string similarity in the word RAM model. We show how to preprocess any string $T \in [0..σ)^n$ in $O(n\logσ/\log n)$ time so that, for any $τ\in[1..n]$, a $τ$-synchronizing set of $T$ can be constructed in $O((n\logτ)/(τ\log n))$ time. Both bounds are optimal in the word RAM model with word size $w=Θ(\log n)$. Previously, the construction time was $O(n/τ)$, either after an $O(n)$-time preprocessing [Kociumaka, Radoszewski, Rytter, Waleń; SICOMP 2024], or without preprocessing if $τ<0.2\log_σn$ [Kempa, Kociumaka; STOC 2019]. A simple version of our method outputs the set as a sorted list in $O(n/τ)$ time, or as a bitmask in $O(n/\log n)$ time. Our optimal construction produces a compact fully indexable dictionary, supporting select queries in $O(1)$ time and rank queries in $O(\log(\tfrac{\logτ}{\log\log n}))$ time, matching unconditional cell-probe lower bounds for $τ\le n^{1-Ω(1)}$. We achieve this via a new framework for processing sparse integer sequences in a custom variable-length encoding. For rank and select queries, we augment the optimal variant of van Emde Boas trees [Pătraşcu, Thorup; STOC 2006] with a deterministic linear-time construction. The above query-time guarantees hold after preprocessing time proportional to the encoding size (in words).

We show that for any unsatisfiable CNF formula $\varphi$ that requires resolution refutation width at least $w$, and for any $1$-stifling gadget $g$ (for example, $g=MAJ_3$), (1) every resolution-over-parities (Res($\oplus$)) refutation of the lifted formula $\varphi \circ g$ of size at most $S$ has depth at least $\Omega(w^2/\log S)$; (2) every Res($\oplus$) refutation of the lifted formula $\varphi \circ g$ has size $\Omega(w^2)$. The first result substantially extends and simplifies all previously known lifting theorems for bounded-depth Res($\oplus$). The lifting result of Itsykson and Knop [IK, ITCS'26] requires gadgets of logarithmic size and applies only to refutations of depth at most $O(n\log n)$, whereas our result applies to nearly quadratic depth. The liftings of Bhattacharya and Chattopadhyay [BC, ECCC'25] and of Byramji and Imagliazzo [BI, arXiv'25] apply to nearly quadratic depth as well, but rely on a much stronger assumption of $(\Omega(n),\Omega(n))$-DT-hardness, which is far less standard than large resolution width. Our proof combines the random-walk-with-restarts method of Alekseev and Itsykson [AI, STOC'25] with a new idea: the random walk is defined relative to the structure of the refutation graph, rather than by a distribution on inputs induced by the formula. Using this technique, we substantially strengthen the supercritical size-depth tradeoff of Itsykson and Knop [IK, ITCS'26], both by improving the depth lower bound and by reducing the size of the separating formulas to polynomial in the number of variables, with the latter resolving an open question. In particular, we construct a family of polynomial-size formulas that admit polynomial-size resolution refutations, while any Res($\oplus$) refutation of depth $o(n^2/\log^4 n)$ necessarily has superpolynomial size. Our second result yields a pure quadratic lower bound on the size of Res($\oplus$) refutations, improving upon the previously known near-quadratic lower bound of [BI, arXiv'25].

The following are prepared remarks that I delivered by Zoom to a student group at my old stomping-grounds of MIT, and which I thought might interest others (even though much of it will be familiar to Shtetl-Optimized regulars). The students asked me to share my “optimistic vision” for the year 2050, so I did my best to oblige. A freewheeling discussion then followed, as a different freewheeling discussion can now follow in the comments section.

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I was asked to share my optimistic vision for the future. The trouble is, optimistic visions for the future are not really my shtick!

It’s not that I’m a miserable, depressed person—I only sometimes am! It’s just that, on a local level, I try to solve the problems in front of me, which have often been problems in computational complexity or quantum computing theory.

And then, on a global level, I worry about the terrifying problems of the world, such as climate change, nuclear war, and of course the resurgence of populist, authoritarian strongmen who’ve turned their backs on the Enlightenment and appeal to the basest instincts of humanity. I won’t name any names.

So then my optimistic vision is simply that we survive all this—“we” meaning the human race, but also meaning communities that I personally care about, like Americans, academics, scientists, and my extended family. We survive all of it so that we can reach the next crisis, the one where we don’t even know what it is yet.

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But I get the sense that you wanted more optimism than that! Since I’ve spent 27 years working in quantum computing, the easiest thing for me to do would be to spin an optimistic story about how QC is going to make our lives so much better in 2050, by, I dunno, solving machine learning and optimization problems much faster, curing cancer, fixing global warming, whatever.

The good news is that there has been spectacular progress over the past couple years toward actually building a scalable QC. We now have two-qubit gates with 99.9% accuracy, close to the threshold where quantum error-correction becomes a net win. We can now do condensed-matter physics simulations that give us numbers that we don’t know how to get classically. I think it’s fair to say that all the key ideas and hardware building blocks for a fault-tolerant quantum computer are now in place, and what remains is “merely” the staggeringly hard engineering problem, which might take a few years, or a decade or more, but should eventually be solved.

The trouble for the optimistic vision is that the applications, where quantum algorithms outperform classical ones, have stubbornly remained pretty specialized. In fact, the two biggest ones remain the two that we knew about in the 1990s:

1. simulation of quantum physics and chemistry themselves, and
2. breaking existing public-key encryption.

Quantum simulation could help with designing better batteries, or solar cells, or high-temperature superconductors, or other materials, but the road from improved understanding to practical value is long and uncertain. Meanwhile, breaking public-key cryptography could help various spy agencies and hackers and criminal syndicates, but it doesn’t obviously help the world.

The quantum speedups that we know outside those two categories—for example, for optimization and machine learning—tend to be either modest or specialized or speculative.

Honestly, the application of QC that excites me the most, by far, is just disproving all the people who said QC was impossible!

So much for QC then.

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And so we come to the elephant in the room—the elephant in pretty much every room nowadays—which is AI. AI has now reached a place that exceeds the imaginations of many of the science-fiction writers of generations past—excelling not only at writing code and solving math competition problems but at depth of emotional understanding. Many of my friends are terrified of where this is leading us—and not in some remote future but in 5 or 10 or 20 years. I think they’re probably correct to be terrified. There’s an enormous range of possible outcomes on the table, including ones where the new superintelligences that we bring into being treat humans basically as humans treated the dodo bird, or the earlier hominids that used to share the earth with us.

But, within this range of outcomes, I think there are also some extremely good ones. Look, for millennia, people have prayed to God or gods for help, life, health, longevity, freedom, justice—and for millennia, God has famously been pretty slow to answer their prayers. A superintelligence that was aligned with human values would be nothing less than a God who did answer, who did deliver all those things, because we had created it to do so. Or for religious people, perhaps such an AI would be the means by which the old God was finally able to deliver all those things into the temporal world. These are the stakes here.

To switch metaphors, people sometimes describe the positive AI-enabled future as “luxury space communism.” AI would take care of all of our material needs, leaving us to seek value in our lives through family, friendships, competition, hobbies, humor, art, entertainment, or exploration. The super-AI would give us the freedom to pursue all those things, but would not give us the freedom to harm each other, to curtail each others’ freedoms, or to build a bad AI capable of overthrowing it. The super-AI would be a singleton, a monotheistic God or its emissary on earth.

Many people say that something would still be missing from this future. After all, we humans would no longer really be needed for anything—for building or advancing or defending civilization. To put a personal fine point on it, my students and colleagues and I wouldn’t needed any more to discover new scientific truths or to write about them. That would all be the AI’s job.

I agree that something would be lost here. But on the other hand, what fraction of us are needed right now for these things? Most humans already derive the meaning in their lives from family and community and enjoying art and music and food and things like that. So maybe the remaining fraction of us should just get over ourselves! On the whole, while this might not be the best future imaginable, I would accept it in a heartbeat given the realistic alternatives on offer. Thanks for listening.

By Scott

July 6-10, 2026 Okinawa, Japan https://www.oist.jp/conference/information-theory-modern-science Submission deadline: May 17, 2026 Registration deadline: June 15, 2026 We are excited to announce the opening of registration for the Information Theory in Modern Science (ITMS) Workshop, hosted at the Okinawa Institute of Science and Technology (OIST), Japan This workshop brings together researchers from information theory, probability, statistics, … Continue reading Information Theory in Modern Science 2026

By shacharlovett

A reader worried about the future.

I am writing this email as a young aspiring researcher/scientist. We live in a period of uncertainty and I have a lot of doubts about the decisions I should make. I've always been interested in mathematics and physics and I believe that a career in this area would be a fulfilling one for me. However, with the development of AI I'm starting to have some worries about my future. It is difficult to understand what is really happening. It feels like everyday these models are improving and sooner rather than later they will render us useless.

I know that the most important objective of the study of mathematics/science is understanding and that AI will not take that pleasure away from us. But still I feel that it will be removing something fundamental to the discipline. If we have a machine that is significantly better than us at solving problems doesn't science lose a part of its magic? The struggle that we face when trying to tackle the genuinely difficult questions is perhaps the most fascinating part of doing research.

I would very much like to read about your opinion on this subject. Will mathematicians/scientists/researchers still have a role to play in all of this? Will science still be an interesting subject to pursue?

There are two questions here, the future of mathematics and the future of mathematicians. Let's take them separately.

Looks like the same letter went to Martin Hairer and highlighted in a recent NYT article about the state of AI doing math and the too-early First Proof project. According to Hairer, "I believe that mathematics is actually quite ‘safe', I haven’t seen any plausible example of an L.L.M. coming up with a genuinely new idea and/or concept."

I don't disagree with Hairer but the state of the art can quickly change. A few months ago I would have said that AI had yet to prove a new theorem, no longer true.

It's too early to tell just how good AI will get in mathematics. Right now it is like an early career graduate student, good at literature search, formalizing proofs, writing paper drafts, exploring known concepts and simple mathematical discussions. But no matter how good it gets, mathematics is a field of infinite concepts, there will always be more to explore as Gödel showed. I do hope AI gets strong at finding new concepts and novel proofs of theorems, and see new math that might not have happened while I'm still here to see it.

For mathematicians the future looks more complicated. AI may never come up with new ideas and Hairer might be right. Or AI could become so good at theorem proving that if you give a conjecture to AI and it can't resolve it, you might as well not try. The true answer is likely in-between and we'll get there slower rather than faster. 

People go into mathematics for different reasons. Some find joy in seeing new and exciting ideas in math. Some like to create good questions and models to help us make sense of mathematical ideas. Some enjoy the struggle of proving new theorems, working against an unseen force that seems to spoil your proofs until you finally break through, and impressing their peers with their prowess. Some take pleasure in education, exciting others in the importance and excitement of mathematics. These will all evolve with advances in AI and the mathematician's role will evolve with it.

My advice: Embrace mathematics research but be ready to pivot as AI evolves. We could be at the precipice of an incredible time for mathematical advances. How can you not be there to see it? And if not, then math needs you even more.

By Lance Fortnow

The hardness vs. randomness paradigm aims to construct pseudorandom generators (PRGs) based on complexity theoretic hardness assumptions. A seminal result in this area is a PRG construction by \cite{NW,IW97}. A sequence of works \cite{KvM,SU01,Umans02,SU05} generalized the result of \cite{NW,IW97} to nondeterministic circuits. More specifically, they showed that if $\E=\DTIME(2^{O(n)})$ requires nondeterministic circuits of size $2^{\Omega(n)}$, then for every sufficiently large $s$, and every $\eps \ge \frac{1}{s}$, there is an $\eps$-PRG $G:\B^{r=O(\log s + \log \frac{1}{\eps})} \to \B^s$ that runs in time $\poly(s)$, and fools size $s$ nondeterministic circuits. In particular, for every size $s$ nondeterministic circuit $C$, \[\Pr[C(G(U_r))=1] \le \Pr[C(U_s)=1] + \eps.\] Applebaum et al. \cite{AASY15} showed that ``black-box techniques'' cannot achieve such results for $\eps=s^{-\omega(1)}$. In order to circumvent this problem, Artemenko et al. \cite{AIKS16} suggested a ``multiplicative'' version of PRGs, which requires that: \[\Pr[C(G(U_r))=1] \le 2 \cdot \Pr[C(U_s)=1] + \eps.\] This still gives that $\Pr[C(G(U_r))=1]$ is very small, if $\Pr[C(U_s)=1]$ is very small, and is therefore suitable for applications that only require this consequence. \cite{AIKS16} constructed such multiplicative PRGs for $\eps=s^{-\omega(1)}$ (based on very strong hardness assumptions). In this paper, we give an optimal construction of multiplicative PRGs for nondeterministic circuits. More specifically, under the same hardness assumption used for (standard) PRGs for nondeterministic circuits, we show that for every $\eps \ge \frac{1}{2^{s}}$, there is a multiplicative PRG $G:\B^{r=O(\log s + \log \frac{1}{\eps})} \to \B^s$ that runs in time $\poly(s)$ and fools size $s$ nondeterministic circuits. This gives the optimal seed length under a hardness assumption that is necessary, and provides improvements in several applications of multiplicative PRGs. Our result improves upon the previous multiplicative PRG construction of \cite{AIKS16}, which uses a stronger hardness assumption against $\Sigma_3$-circuits, and where the seed length is the suboptimal $r=O(\log s) + O(\log \frac{1}{\eps})^2$. Our result also improves upon the recent multiplicative PRG of Shaltiel \cite{ShaltielCCC25} that only achieves very small stretch (the output length in \cite{ShaltielCCC25} is less than twice the seed length). \smallskip We also show that black-box techniques cannot give a version of our result where ``nondeterministic'' is replaced by ``deterministic''. This justifies the current situation where hardness for nondeterministic circuits is used even if one only wants low error multiplicative PRGs that fool \emph{deterministic} circuits. \smallskip Our PRG construction borrows ideas from the recent ``low stretch'' PRG of Shaltiel \cite{ShaltielCCC25}, and the (standard) PRG construction of Shaltiel and Umans \cite{SU01}. Loosely speaking, we aim to get the ``multiplicativity'' of the former, and the ``large stretch'' of the latter. While both approaches generalize the list-decoding results of Sudan, Trevisan and Vadhan \cite{STV}, the two results are tailored to two very different parameter regimes, and we introduce several new ideas to make the two approaches co-exist.

Authors: Changryeol Lee

While prior work established a verifier-based polynomial time framework for NP, explicit deterministic machines for concrete NP-complete problems have remained elusive. In this paper, we construct fully specified deterministic Turing Machines (DTMs) for SAT and Subset-Sum within a polynomial-time NP verifier simulation framework. We show that both machines operate in polynomial time and, for satisfiable instances, deterministically generate valid witnesses, thereby extending the framework to deterministic FNP computation without increasing the degree of polynomial complexity. Furthermore, we provide a complete implementation of the framework, including the dynamic computation graph, feasible-graph construction, verification walks, and Turing-machine simulation via edge extensions. The implementation behaves in accordance with the predicted polynomial-time bounds. To ensure transparency and reproducibility, the complete Python implementation and source code are made available in a public online repository.

Authors: Katarína Cechlárová, Ildikó Schlotter

Consider an election where the set of candidates is partitioned into parties, and each party must choose exactly one candidate to nominate for the election held over all nominees. The Necessary President problem asks whether a candidate, if nominated, becomes the winner of the election for all possible nominations from other parties. We study the computational complexity of Necessary President for several voting rules. We show that while this problem is solvable in polynomial time for Borda, Maximin, and Copeland$^α$ for every $α\in [0,1]$, it is $\mathsf{coNP}$-complete for general classes of positional scoring rules that include $\ell$-Approval and $\ell$-Veto, even when the maximum size of a party is two. For such positional scoring rules, we show that Necessary President is $\mathsf{W}[2]$-hard when parameterized by the number of parties, but fixed-parameter tractable with respect to the number of voter types. Additionally, we prove that Necessary President for Ranked Pairs is $\mathsf{coNP}$-complete even for maximum party size two, and $\mathsf{W}[1]$-hard with respect to the number of parties; remarkably, both of these results hold even for constant number of voters.

Authors: Colin Cleveland, Bart de Keijzer, Maria Polukarov

We study the computational complexity of strategic behaviour in primary elections. Unlike direct voting systems, primaries introduce a multi-stage process in which voters first influence intra-party nominees before a general election determines the final winner. While previous work has evaluated primaries via welfare distortion, we instead examine their game-theoretic properties. We formalise a model of primaries under first-past-the-post with fixed tie-breaking and analyse voters' strategic behaviour. We show that determining whether a pure Nash equilibrium exists is $Σ_2^{\mathbf P}$-complete, computing a best response is NP-complete, and deciding the existence of subgame-perfect equilibria in sequential primaries is PSPACE-complete. These results reveal that primaries fundamentally increase the computational difficulty of strategic reasoning, situating them as a rich source of complexity-theoretic challenges within computational social choice.

Authors: Michaela Borzechowski, Sebastian Haslebacher, Hung P. Hoang, Patrick Schnider, Simon Weber

The famous Ham-Sandwich theorem states that any $d$ point sets in $\mathbb{R}^d$ can be simultaneously bisected by a single hyperplane. The $α$-Ham-Sandwich theorem gives a sufficient condition for the existence of biased cuts, i.e., hyperplanes that do not cut off half but some prescribed fraction of each point set. We give two new proofs for this theorem. The first proof is completely combinatorial and highlights a strong connection between the $α$-Ham-Sandwich theorem and Unique Sink Orientations of grids. The second proof uses point-hyperplane duality and the Poincaré-Miranda theorem and allows us to generalize the result to and beyond oriented matroids. For this we introduce a new concept of rainbow arrangements, generalizing colored pseudo-hyperplane arrangements. Along the way, we also show that the realizability problem for rainbow arrangements is $\exists \mathbb{R}$-complete, which also implies that the realizability problem for grid Unique Sink Orientations is $\exists \mathbb{R}$-complete.

Authors: Palash Dey, Anubhav Dhar, Ashlesha Hota, Sudeshna Kolay, Aritra Mitra

In this paper, we study the Maximum Common Vertex Subgraph problem: Given two input graphs $G_1,G_2$ and a non-negative integer $h$, is there a common subgraph $H$ on at least $h$ vertices such that there is no isolated vertex in $H$. In other words, each connected component of $H$ has at least $2$ vertices. This problem naturally arises in graph theory along with other variants of the well-studied Maximum Common Subgraph problem and also has applications in computational social choice. We show that this problem is NP-hard and provide an FPT algorithm when parameterized by $h$. Next, we conduct a study of the problem on common structural parameters like vertex cover number, maximum degree, treedepth, pathwidth and treewidth of one or both input graphs. We derive a complete dichotomy of parameterized results for our problem with respect to individual parameterizations as well as combinations of parameterizations from the above structural parameters. This provides us with a deep insight into the complexity theoretic and parameterized landscape of this problem.

Authors: Guangyan Zhou

We study the algorithmic decidability of the domination number in the Erdos-Renyi random graph model $G(n,p)$. We show that for a carefully chosen edge probability $p=p(n)$, the domination problem exhibits a strong irreducible property. Specifically, for any constant $0

Authors: Joshua Brakensiek, Lorenzo Ciardo, Venkatesan Guruswami, Aaron Potechin, Stanislav Živný

In this paper, we continue the study of robust satisfiability of promise CSPs (PCSPs), initiated in (Brakensiek, Guruswami, Sandeep, STOC 2023 / Discrete Analysis 2025), and obtain the following results: For the PCSP 1-in-3-SAT vs NAE-SAT with negations, we prove that it is hard, under the Unique Games conjecture (UGC), to satisfy $1-Ω(1/\log (1/ε))$ constraints in a $(1-ε)$-satisfiable instance. This shows that the exponential loss incurred by the BGS algorithm for the case of Alternating-Threshold polymorphisms is necessary, in contrast to the polynomial loss achievable for Majority polymorphisms. For any Boolean PCSP that admits Majority polymorphisms, we give an algorithm satisfying $1-O(\sqrtε)$ fraction of the weaker constraints when promised the existence of an assignment satisfying $1-ε$ fraction of the stronger constraints. This significantly generalizes the Charikar--Makarychev--Makarychev algorithm for 2-SAT, and matches the optimal trade-off possible under the UGC. The algorithm also extends, with the loss of an extra $\log (1/ε)$ factor, to PCSPs on larger domains with a certain structural condition, which is implied by, e.g., a family of Plurality polymorphisms. We prove that assuming the UGC, robust satisfiability is preserved under the addition of equality constraints. As a consequence, we can extend the rich algebraic techniques for decision/search PCSPs to robust PCSPs. The methods involve the development of a correlated and robust version of the general SDP rounding algorithm for CSPs due to (Brown-Cohen, Raghavendra, ICALP 2016), which might be of independent interest.

Authors: Xi Chen, Yuhao Li, Mihalis Yannakakis

We study the problem of finding an $ε$-fixed point of a contraction map $f:[0,1]^k\mapsto[0,1]^k$ under both the $\ell_\infty$-norm and the $\ell_1$-norm. For both norms, we give an algorithm with running time $O(\log^{\lceil k/2\rceil}(1/ε))$, for any constant $k$. These improve upon the previous best $O(\log^k(1/ε))$-time algorithm for the $\ell_{\infty}$-norm by Shellman and Sikorski [SS03], and the previous best $O(\log^k (1/ε))$-time algorithm for the $\ell_{1}$-norm by Fearnley, Gordon, Mehta and Savani [FGMS20].

Authors: Jean Cardinal, Micha Sharir

Semialgebraic graphs are graphs whose vertices are points in $\mathbb{R}^d$, and adjacency between two vertices is determined by the truth value of a semialgebraic predicate of constant complexity. We show how to harness polynomial partitioning methods to construct compact adjacency labeling schemes for families of semialgebraic graphs. That is, we show that for any family of semialgebraic graphs, given a graph on $n$ vertices in this family, we can assign a label consisting of $O(n^{1-2/(d+1) + \varepsilon})$ bits to each vertex (where $\varepsilon > 0$ can be made arbitrarily small and the constant of proportionality depends on $\varepsilon$ and on the complexity of the adjacency-defining predicate), such that adjacency between two vertices can be determined solely from their two labels, without any additional information. We obtain for instance that unit disk graphs and segment intersection graphs have such labelings with labels of $O(n^{1/3 + \varepsilon})$ bits. This is in contrast to their natural implicit representation consisting of the coordinates of the disk centers or segment endpoints, which sometimes require exponentially many bits. It also improves on the best known bound of $O(n^{1-1/d}\log n)$ for $d$-dimensional semialgebraic families due to Alon (Discrete Comput. Geom., 2024), a bound that holds more generally for graphs with shattering functions bounded by a degree-$d$ polynomial. We also give new bounds on the size of adjacency labels for other families of graphs. In particular, we consider semilinear graphs, which are semialgebraic graphs in which the predicate only involves linear polynomials. We show that semilinear graphs have adjacency labels of size $O(\log n)$. We also prove that polygon visibility graphs, which are not semialgebraic in the above sense, have adjacency labels of size $O(\log^3 n)$.

Authors: Nathaniel K. Brown, Ben Langmead

The move structure represents permutations with long contiguously permuted intervals in compressed space with optimal query time. They have become an important feature of compressed text indexes using space proportional to the number of Burrows-Wheeler Transform (BWT) runs, often applied in genomics. This is in thanks not only to theoretical improvements over past approaches, but great cache efficiency and average case query time in practice. This is true even without using the worst case guarantees provided by the interval splitting balancing of the original result. In this paper, we show that an even simpler type of splitting, length capping by truncating long intervals, bounds the average move structure query time to optimal whilst obtaining a superior construction time than the traditional approach. This also proves constant query time when amortized over a full traversal of a single cycle permutation from an arbitrary starting position. Such a scheme has surprising benefits both in theory and practice. We leverage the approach to improve the representation of any move structure with $r$ runs over a domain $n$ to $O(r \log r + r \log \frac{n}{r})$-bits of space. The worst case query time is also improved to $O(\log \frac{n}{r})$ without balancing. An $O(r)$-time and $O(r)$-space construction lets us apply the method to run-length encoded BWT (RLBWT) permutations such as LF and $φ$ to obtain optimal-time algorithms for BWT inversion and suffix array (SA) enumeration in $O(r)$ additional working space. Finally, we provide the RunPerm library, providing flexible plug and play move structure support, and use it to evaluate our splitting approach. Experiments find length capping results in faster move structures, but also a space reduction: at least $\sim 40\%$ for LF across large repetitive genomic collections.

Authors: Anouk Duyster, Tomasz Kociumaka

A Random Access query to a string $T\in [0..σ)^n$ asks for the character $T[i]$ at a given position $i\in [0..n)$. In $O(n\logσ)$ bits of space, this fundamental task admits constant-time queries. While this is optimal in the worst case, much research has focused on compressible strings, hoping for smaller data structures that still admit efficient queries. We investigate the grammar-compressed setting, where $T$ is represented by a straight-line grammar. Our main result is a general trade-off that optimizes Random Access time as a function of string length $n$, grammar size (the total length of productions) $g$, alphabet size $σ$, data structure size $M$, and word size $w=Ω(\log n)$ of the word RAM model. For any $M$ with $g\log n 0$ [Belazzougui et al.; ESA'15], [Ganardi, Jeż, Lohrey; J. ACM 2021]. The only tight lower bound [Verbin and Yu; CPM'13] was $Ω(\frac{\log n}{\log\log n})$ for $w=Θ(\log n)$, $n^{Ω(1)}\le g\le n^{1-Ω(1)}$, and $M=g\log^{Θ(1)}n$. In contrast, our result yields tight bounds in all relevant parameters and almost all regimes. Our data structure admits efficient deterministic construction. It relies on novel grammar transformations that generalize contracting grammars [Ganardi; ESA'21]. Beyond Random Access, its variants support substring extraction, rank, and select.

Authors: Frederik Glitzner

Stable matching is a fundamental area with many practical applications, such as centralised clearinghouses for school choice or job markets. Recent work has introduced the paradigm of near-feasibility in capacitated matching settings, where agent capacities are slightly modified to ensure the existence of desirable outcomes. While useful when no stable matching exists, or some agents are left unmatched, it has not previously been investigated whether near-feasible stable matchings satisfy desirable properties with regard to their stability in the original instance. Furthermore, prior works often leave open deviation incentive issues that arise when the centralised authority modifies agents' capacities. We consider these issues in the Stable Fixtures problem model, which generalises many classical models through non-bipartite preferences and capacitated agents. We develop a formal framework to analyse and quantify agent incentives to adhere to computed matchings. Then, we embed near-feasible stable matchings in this framework and study the trade-offs between instability, capacity modifications, and computational complexity. We prove that capacity modifications can be simultaneously optimal at individual and aggregate levels, and provide efficient algorithms to compute them. We show that different modification strategies significantly affect stability, and establish that minimal modifications and minimal deviation incentives are compatible and efficiently computable under general conditions. Finally, we provide exact algorithms and experimental results for tractable and intractable versions of these problems.

Authors: Christian Bertram, Mads Vestergaard Jensen

Given an undirected graph $G=(V, E)$, the Personalized PageRank (PPR) of $t\in V$ with respect to $s\in V$, denoted $π(s,t)$, is the probability that an $α$-discounted random walk starting at $s$ terminates at $t$. We study the time complexity of estimating $π(s,t)$ with constant relative error and constant failure probability, whenever $π(s,t)$ is above a given threshold parameter $δ\in(0,1)$. We consider common graph-access models and furthermore study the single source, single target, and single node (PageRank centrality) variants of the problem. We provide a complete characterization of PPR estimation in undirected graphs by giving tight bounds (up to logarithmic factors) for all problems and model variants in both the worst-case and average-case setting. This includes both new upper and lower bounds. Tight bounds were recently obtained by Bertram, Jensen, Thorup, Wang, and Yan for directed graphs. However, their lower bound constructions rely on asymmetry and therefore do not carry over to undirected graphs. At the same time, undirected graphs exhibit additional structure that can be exploited algorithmically. Our results resolve the undirected case by developing new techniques that capture both aspects, yielding tight bounds.

Authors: Ernestine Großmann, Christian Schulz, Darren Strash, Antonie Wagner

This work addresses the well-known Maximum Independent Set problem in the context of hypergraphs. While this problem has been extensively studied on graphs, we focus on its strong extension to hypergraphs, where edges may connect any number of vertices. A set of vertices in a hypergraph is strongly independent if there is at most one vertex per edge in the set. One application for this problem is to find perfect minimal hash functions. We propose nine new data reduction rules specifically designed for this problem. Our reduction routine can serve as a preprocessing step for any solver. We analyze the impact on the size of the reduced instances and the performance of several subsequent solvers when combined with this preprocessing. Our results demonstrate a significant reduction in instance size and improvements in running time for subsequent solvers. The preprocessing routine reduces instances, on average, to 22% of their original size in 6.76 seconds. When combining our reduction preprocessing with the best-performing exact solver, we observe an average speedup of 3.84x over not using the reduction rules. In some cases, we can achieve speedups of up to 53x. Additionally, one more instance becomes solvable by a method when combined with our preprocessing.

Authors: Bernhard Haeupler, Antti Roeyskoe, Zhijun Zhang

All parallel algorithms for directed connectivity and shortest paths crucially rely on efficient shortcut constructions that add a linear number of transitive closure edges to a given DAG to reduce its diameter. A long sequence of works has studied both (efficient) shortcut constructions and impossibility results on the best diameter and therefore the best parallelism that can be achieved with this approach. This paper introduces a new conceptual and technical tool, called certified shortcuts, for this line of research in the form of a simple and natural structural criterion that holds for any shortcut constructed by an efficient (combinatorial) algorithm. It allows us to drastically simplify and strengthen existing impossibility results by proving that any near-linear-time shortcut-based algorithm cannot reduce a graph's diameter below $n^{1/4-o(1)}$. This greatly improves over the $n^{2/9-o(1)}$ lower bound of [HXX25] and seems to be the best bound one can hope for with current techniques. Our structural criterion also precisely captures the constructiveness of all known shortcut constructions: we show that existing constructions satisfy the criterion if and only if they have known efficient algorithms. We believe our new criterion and perspective of looking for certified shortcuts can provide crucial guidance for designing efficient shortcut constructions in the future.

Authors: Vijay K. Garg, Rohan Garg

We present methods to compute least fixed points of multiple monotone inflationary functions in parallel and distributed settings. While the classic Knaster-Tarski theorem addresses a single function with sequential iteration, modern computing systems require parallel execution with overwrite semantics, non-atomic updates, and stale reads. We prove three convergence theorems under progressively relaxed synchronization: (1) Interleaving semantics with fair scheduling, (2) Parallel execution with update-only-on-change semantics (processes write only on those coordinates whose values change), and (3) Distributed execution with bounded staleness (updates propagate within $T$ rounds) and $i$-locality (each process modifies only its own component). Our approach differs from prior work in fundamental ways: Cousot-Cousot's chaotic iteration uses join-based merges that preserve information. Instead, we use coordinate-wise overwriting. Bertsekas's asynchronous methods assume contractions. We use coordinate-wise overwriting with structural constraints (locality, bounded staleness) instead. Applications include parallel and distributed algorithms for the transitive closure, stable marriage, shortest paths, and fair division with subsidy problems. Our results provide the first exact least-fixed-point convergence guarantees for overwrite-based parallel updates without join operations or contraction assumptions.

Authors: Kishen N Gowda, Willem Fletcher, MohammadHossein Bateni, Laxman Dhulipala, D Ellis Hershkowitz, Rajesh Jayaram, Jakub Łącki

Hierarchical Agglomerative Clustering (HAC) is a widely-used clustering method based on repeatedly merging the closest pair of clusters, where inter-cluster distances are determined by a linkage function. Unlike many clustering methods, HAC does not optimize a single explicit global objective; clustering quality is therefore primarily evaluated empirically, and the choice of linkage function plays a crucial role in practice. However, popular classical linkages, such as single-linkage, average-linkage and Ward's method show high variability across real-world datasets and do not consistently produce high-quality clusterings in practice. In this paper, we propose \emph{Chamfer-linkage}, a novel linkage function that measures the distance between clusters using the Chamfer distance, a popular notion of distance between point-clouds in machine learning and computer vision. We argue that Chamfer-linkage satisfies desirable concept representation properties that other popular measures struggle to satisfy. Theoretically, we show that Chamfer-linkage HAC can be implemented in $O(n^2)$ time, matching the efficiency of classical linkage functions. Experimentally, we find that Chamfer-linkage consistently yields higher-quality clusterings than classical linkages such as average-linkage and Ward's method across a diverse collection of datasets. Our results establish Chamfer-linkage as a practical drop-in replacement for classical linkage functions, broadening the toolkit for hierarchical clustering in both theory and practice.

Authors: Anders Aamand, Justin Y. Chen, Sandeep Silwal

We study differentially private continual release of the number of distinct items in a turnstile stream, where items may be both inserted and deleted. A recent work of Jain, Kalemaj, Raskhodnikova, Sivakumar, and Smith (NeurIPS '23) shows that for streams of length $T$, polynomial additive error of $Ω(T^{1/4})$ is necessary, even without any space restrictions. We show that this additive error lower bound can be circumvented if the algorithm is allowed to output estimates with both additive \emph{and multiplicative} error. We give an algorithm for the continual release of the number of distinct elements with $\text{polylog} (T)$ multiplicative and $\text{polylog}(T)$ additive error. We also show a qualitatively similar phenomenon for estimating the $F_2$ moment of a turnstile stream, where we can obtain $1+o(1)$ multiplicative and $\text{polylog} (T)$ additive error. Both results can be achieved using polylogarithmic space whereas prior approaches use polynomial space. In the sublinear space regime, some multiplicative error is necessary even if privacy is not a consideration. We raise several open questions aimed at better understanding trade-offs between multiplicative and additive error in private continual release.

Authors: Mateusz Basiak, Marcin Bienkowski, Guy Even, Agnieszka Tatarczuk

The online bisection problem requires maintaining a dynamic partition of $n$ nodes into two equal-sized clusters. Requests arrive sequentially as node pairs. If the nodes lie in different clusters, the algorithm pays unit cost. After each request, the algorithm may migrate nodes between clusters at unit cost per node. This problem models datacenter resource allocation where virtual machines must be assigned to servers, balancing communication costs against migration overhead. We study the variant where requests are restricted to edges of a ring network, an abstraction of ring-allreduce patterns in distributed machine learning. Despite this restriction, the problem remains challenging with an $Ω(n)$ deterministic lower bound. We present a randomized algorithm achieving $O(\varepsilon^{-3} \cdot \log^2 n)$ competitive ratio using resource augmentation that allows clusters of size at most $(3/4 + \varepsilon) \cdot n$. Our approach formulates the problem as a metrical task system with a restricted state space. By limiting the number of cut-edges (i.e., ring edges between clusters) to at most $2k$, where $k = Θ(1/\varepsilon)$, we reduce the state space from exponential to polynomial (i.e., $n^{O(k)}$). The key technical contribution is proving that this restriction increases cost by only a factor of $O(k)$. Our algorithm follows by applying the randomized MTS solution of Bubeck et al. [SODA 2019]. The best result to date for bisection with ring demands is the $O(n \cdot \log n)$-competitive deterministic online algorithm of Rajaraman and Wasim [ESA 2024] for the general setting. While prior work for ring-demands by Räcke et al. [SPAA 2023] achieved $O(\log^3 n)$ for multiple clusters, their approach employs a resource augmentation factor of $2+\varepsilon$, making it inapplicable to bisection.

Authors: Robert Ganian, Viktoriia Korchemna

We investigate the parameterized complexity of Bayesian Network Structure Learning (BNSL), a classical problem that has received significant attention in empirical but also purely theoretical studies. We follow up on previous works that have analyzed the complexity of BNSL w.r.t. the so-called superstructure of the input. While known results imply that BNSL is unlikely to be fixed-parameter tractable even when parameterized by the size of a vertex cover in the superstructure, here we show that a different kind of parameterization - notably by the size of a feedback edge set - yields fixed-parameter tractability. We proceed by showing that this result can be strengthened to a localized version of the feedback edge set, and provide corresponding lower bounds that complement previous results to provide a complexity classification of BNSL w.r.t. virtually all well-studied graph parameters. We then analyze how the complexity of BNSL depends on the representation of the input. In particular, while the bulk of past theoretical work on the topic assumed the use of the so-called non-zero representation, here we prove that if an additive representation can be used instead then BNSL becomes fixed-parameter tractable even under significantly milder restrictions to the superstructure, notably when parameterized by the treewidth alone. Last but not least, we show how our results can be extended to the closely related problem of Polytree Learning.

Authors: Julia Gaudio, Charlie K. Guan

Consider $n$ points generated uniformly at random in the unit square, and let $L_n$ be the length of their optimal traveling salesman tour. Beardwood, Halton, and Hammersley (1959) showed $L_n / \sqrt n \to β$ almost surely as $n\to \infty$ for some constant $β$. The exact value of $β$ is unknown but estimated to be approximately $0.71$ (Applegate, Bixby, Chvátal, Cook 2011). Beardwood et al. further showed that $0.625 \leq β\leq 0.92116.$ Currently, the best known bounds are $0.6277 \leq β\leq 0.90380$, due to Gaudio and Jaillet (2019) and Carlsson and Yu (2023), respectively. The upper bound was derived using a computer-aided approach that is amenable to lower bounds with improved computation speed. In this paper, we show via simulation and concentration analysis that future improvement of the $0.90380$ is limited to $\sim0.88$. Moreover, we provide an alternative tour-constructing heuristic that, via simulation, could potentially improve the upper bound to $\sim0.85$. Our approach builds on a prior \emph{band-traversal} strategy, initially proposed by Beardwood et al. (1959) and subsequently refined by Carlsson and Yu (2023): divide the unit square into bands of height $Θ(1/\sqrt{n})$, construct paths within each band, and then connect the paths to create a TSP tour. Our approach allows paths to cross bands, and takes advantage of pairs of points in adjacent bands which are close to each other. A rigorous numerical analysis improves the upper bound to $0.90367$.

The Faculty of Science at the University of Calgary invites applications for a Tenure-Track Assistant Professor in Computer Science position with a focus on Theoretical Foundations of Computer Science.

Website: https://careers.ucalgary.ca/jobs/17169318-tenure-track-assistant-professor-research-in-computer-science-faculty-of-science
Email: cpsc.hiring@ucalgary.ca

By shacharlovett

This is a live blog of Lecture 3 of my graduate seminar “Feedback, Learning, and Adaptation.” A table of contents is here.

I could teach an entire semester-long graduate class on gradient descent. First, I’d present gradient descent. Then I’d move to accelerated gradient descent. Then I could teach stochastic gradient descent, coordinate descent, projected gradient descent, proximal gradient descent… This would get us to Spring break. We could wrap up the semester with other assorted gradient potpourri. Indeed, Steve Wright and I packaged this course into a textbook: Optimization for Data Analysis.

Steve and I were inspired by the thousands of machine learning and optimization papers of the 2010s that made minor extensions in this gradient zoo. All of these papers proved their methods worked in the same way. They set up a Lyapunov function. They showed that it decreased as the algorithm evolved. QED.

Those Lyapunov functions were sometimes easy to come by. You’d always start with the function value itself. If it significantly decreased every iteration, then the algorithm would converge. You could also study the distance of the current iterate to the optimal solution. It took me a decade of beating my head against Nesterov’s inscrutable estimate sequences to realize that he was actually using a Lyapunov function too. In Nesterov’s accelerated method, this Lyapunov function has the form:

Showing functions like this were Lyapunov functions required pages of calculations, but all of these were manipulating exactly two inequalities. The most important assumption when analyzing gradient descent is that the gradients are Lipschitz. This means that the slope of the gradient function is bounded. Oftentimes, we also assume that the functions are strongly convex. This is equivalent to assuming the slope of the gradient function is bounded below.

Together, we had that the following two inequalities were true for any x and z.

Here, L is the Lipschitz constant of the gradient. m is the strong convexity parameter. Sometimes we’d use the second inequality with m=0. You might call those functions weakly convex. Convergence proofs cleverly sum up these two inequalities evaluated at different points in space to show that some Lyapunov function decreases. After enough Tetris-like puzzling, you surely prove that the Lyapunov function decreases.

These analyses appear to be assessing the stability of a dynamical system. That’s because they are. Gradient methods control a nonlinear system that takes a vector as input and outputs the gradient of a convex function evaluated at that vector.

The algorithm feeds “x” into the black box. The black box spits out “g.” The algorithm does some thinking and spits out another x. Eventually, the g emitted by the black box is always equal to zero.

In fact, all of the gradient-based algorithms are equivalent to PID controllers. Gradient descent is literally an integral controller. It is even after the same goals: gradient descent wants to find a point where the derivative is zero. Integral control seeks zero steady-state error. Accelerated methods are PID controllers. Projected gradient is a PI controller.

What if we just relabel that picture to align with control theory notation:

The slope bound assumption on the gradients is equivalent to assuming the black box has gain bounded between an upper and lower bound. This is the sort of thing control theorists have studied for a century. They call such nonlinear functions “sector bounded” and have a variety of tools to verify control algorithms when such uncertain nonlinearities are in the feedback loop.

In the paper “Analysis of Optimization Algorithms by Integral Quadratic Constraints,” Laurent Lessard, Andy Packard, and I translated these techniques to optimization algorithms. This let us search for Lyapunov-like proofs that your algorithm converges. With these tools, we could derive the same convergence rates and get novel robustness arguments. And the analyses were automatable, in the sense that we derived our proofs using other optimization algorithms.

A complementary approach to this problem developed by optimizers is the PEP framework, which uses a language more native to optimization. Both proof systems work because positive linear combinations of positive things are positive. Thus, you try to show a Lyapunov function decreases by writing this statement as an equality, and showing it’s the linear combination of a bunch of these inequalities. The computer can do this for you.

Lots of interesting insights come from this parallel between optimization and control. For instance, it shows there is no way to “fix” simple momentum methods like the Heavy Ball Method to prove they converge globally. The automated proof framework also helped us identify perturbations to the methods that could sometimes yield faster convergence rates or greater numerical stability.

But one thing I haven’t been able to do is turn this around: I want to teach a semester-long graduate course on PID control that would feel like my optimization class. I was hoping to get a start during this graduate seminar. I wanted to make it clear that most of the analysis could be done by Lyapunov functions. I wanted to move beyond sector-bounded nonlinear maps to more common dynamical system models in which people apply PID controllers. And I want to do all of this without ever taking a Laplace transform.

If there are any control theorists out there reading this with ideas for putting such a course together, please let me know! I know many would argue that PID controllers are solved, and the interesting stuff happens at a higher level. But to push the limits of what modern learning-control systems do, we have to understand the PID controls at the innermost loops of the complex system. Explaining this part of the architecture in a clean, modern way is a good pedagogical challenge for my control theory friends.

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By Ben Recht

The Algorithms group at IISc Bengaluru invites posdoc applications. Areas include Approximation/Online Algorithms, Game Theory, Computational Geometry, Optimization, Learning, and more. Fellowship: ₹80,000–1,30,000/month + travel/research grant.

Faculty: Siddharth Barman, Arindam Khan, Anand Louis, Rahul Saladi.

𝗔𝗽𝗽𝗹𝗶𝗰𝗮𝘁𝗶𝗼𝗻 𝗟𝗶𝗻𝗸: https://forms.gle/moz2vx7tiNFCbXmS6

Website: https://www.linkedin.com/posts/arindam-khan-445ab615_postdoc-jobs-algorithms-activity-7427285529886580736-6tUk?utm_source=share&utm_medium=member_desktop&rcm=ACoAAAMwLfUBJutL4b0gGJNLzdNG9Zwai-rt7_M
Email: ARINDAMKHAN@IISC.AC.IN

By shacharlovett

Authors: Ismael Rodriguez

We identify a few conditions $X$ such that $(P=NP \wedge X) \;\Rightarrow\; P=PSPACE$.