Theory of Computing Report

Authors: Christopher Williamson

A pair of probability distributions over $\{0,1\}^n$ is said to be $(k,δ)$-wise indistinguishable if all of the size $k$ marginals are within statistical distance at most $δ$. Previous works introduced this concept and study when and how well one can distinguish between such a pair of symmetric distributions by observing $t$ bits. We use a simple hypergeometric smoothing approach and Hahn polynomials to obtain new upper bounds that apply across a wider range of parameters and improve previously available bounds in several regimes. In particular, prior works left open the basic question of whether there exist constants $0

Authors: Yuetai Li, Fengqing Jiang, Yichen Feng, Kaiyuan Zheng, Luyao Niu, Bhaskar Ramasubramanian, Basel Alomair, Linda Bushnell, Radha Poovendran

Many online decision problems over combinatorial actions are addressed via convex relaxations, leading to online convex optimization with piecewise linear objectives and induced polyhedral structure. We show that regret in such problems is governed by \emph{polyhedral instability}: the number of changes of the active region. Under full information feedback and fixed partition assumptions, if $\mathrm{RS}_T$ denotes the number of region switches and $V_{\max}$ the maximum number of vertices per region, we prove $\Regret_T= Θ(\sqrt{(1+\mathrm{RS}_T)\,T\,\log V_{\max}})$ interpolating between experts-like and dimension-dependent OCO rates. For online submodular--concave games under Lovász convexification, this reduces to the permutation-switch count $\mathrm{SC}_T$, yielding the matching rate $\Regret_T= Θ(\sqrt{(1+\mathrm{SC}_T)\,T\,\log n})$. Experiments on synthetic and real combinatorial problems (shortest path, influence maximization) validate the predicted scaling and indicate that low-instability regimes can arise in practice without explicit enumeration of actions.

Authors: Tatiana Rocha Avila, Julian Christoph Brinkmann, Alexander Leonhardt, Conrad Schecker

We consider the Minimum-$(k,ρ)$-$\mathrm{Shortcut}$ problem ($\min(k,ρ)\text{-}\mathrm{Shortcut}$), where the goal is to find the smallest set of shortcut edges such that every vertex in a given graph can reach its $ρ$ closest vertices using paths of at most $k$ edges. This is a fundamental graph optimization problem used to accelerate parallel shortest path algorithms. It is well-known that the problem is trivially solvable for the cases $k=1$ and $k\geqρ$. While recent work by Leonhardt, Meyer, and Penschuck (ESA 2024) showed that in undirected graphs $\min(k,ρ)\text{-}\mathrm{Shortcut}$ is NP-hard for $k\geq 3$ if $ρ=Θ(n^ε)$, the boundary where the problem transitions from polynomial-time solvable to NP-hard remained open. In this paper, we narrow this gap significantly. We present a simpler and more direct reduction from the Hitting Set problem which establishes that $\min(k,ρ)\text{-}\mathrm{Shortcut}$ is NP-hard for $k\geq2$ and $ρ\geq k+2$ in both directed and undirected graphs. Complementing this, we use the symmetry of the undirected case to show that $ρ=k+1$ is solvable in polynomial time, a regime where the directed version remains a candidate for NP-hardness. Therefore, we obtain an almost complete characterization of the complexity of $\min(k,ρ)\text{-}\mathrm{Shortcut}$, with the sole remaining open case being $ρ= k+1$ in the directed setting.

Authors: Johannes K. Fichte, Markus Hecher, Yasir Mahmood, Zhengjun Wang

Argumentation is an important topic of AI for modelling and reasoning about arguments. In abstract argumentation, we consider directed graphs, so-called argumentation frameworks (AF), that express conflicts between arguments. The semantics is defined by the notion of extensions, which are sets of arguments that satisfy particular relationship conditions in the AF. Usually, standard reasoning in argumentation do not reveal how far apart extensions are. We introduce a quantitative notion of diversity of extensions based on the symmetric difference and provide a systematic complexity classification. Intuitively, diversity captures whether extensions of a framework (accepted viewpoints) differ only marginally or represent fundamentally incompatible sets of arguments. We study whether an AF admits k-diverse extensions, admits k-diverse extensions covering specific arguments, and to compute the largest k for which an AF admits k-diverse extensions. We outline a prototype and provide an evaluation for computing diversity levels.

Authors: Arshia Soltani Moakahr, Faraz Ghahremani, Kiarash Banihashem, MohammadTaghi Hajiaghayi

Decision tree learning has long been a central topic in theoretical computer science, driven by its practical importance. A fundamental and widely used method for decision tree construction is the top-down greedy heuristic, which recursively splits on the most influential variable. Despite its empirical success, theoretical analysis of this heuristic has been limited. A recent breakthrough by Blanc et al. (ITCS, 2020) provided the first rigorous theoretical guarantees for the greedy approach, but only under the uniform distribution. We extend this analysis to the more general and practically relevant setting of arbitrary product distributions. Our main result shows that for any function $f$ computable by an optimal decision tree of size $s$, maximum depth $D_{\text{opt}}$, and average depth $Δ_{\text{opt}}$, the greedy heuristic constructs an $ε$-approximating tree whose size grows at most with $\exp\bigl(Δ_{\text{opt}} D_{\text{opt}} \log(e/ε)\bigr)$. In the special case where the optimal tree is a full binary tree, this bound improves upon the bound of Blanc et al. and holds under a strictly broader class of distributions. Moreover, we present an algorithm based on the top-down greedy heuristic that is entirely parameter-free -- it requires no prior knowledge of the optimal tree's size or depth -- offering a practical advantage over Blanc et al.'s method.

Authors: Nathaniel Glover, Jan Hoffmann

Hofmann (1999) introduced the functional programming language LFPL to characterize the functions computable in polynomial time using an affine type system. LFPL enables a natural programming style, including nested recursion, and has inspired the development of type systems for automatic cost analysis, linear dependent type theories, and efficient memory management in functional programming languages. Despite its prominence, there does not exist a self-contained presentation, let alone a full mechanization, of LFPL and its core metatheory. This article presents a modern account and mechanization of LFPL and its metatheory with the goal of being self-contained and accessible while streamlining the strongest-known soundness and completeness results. The soundness proof works with the language LFPL+, which extends LFPL with additional language features. The proof is novel, adapting a technique by Aehlig and Schwichtenberg (2002) to construct explicit polynomials that bound the cost of an LFPL+ expression with respect to a big-step cost semantics. The completeness proof shows that LFPL programs can simulate polynomial-time Turing machines while only relying on restricted forms of linear functions and lists. It has the same structure as the original proof by Hofmann (2002) but greatly simplifies the core argument with a novel stack-like data structure that is implemented with first-class functions and lists. The mechanization includes the full soundness and completeness proofs, and serves as one of the first case studies of mechanized metatheory in the recently developed proof assistant Istari.

Authors: Alkida Balliu, Sebastian Brandt, Fabian Kuhn, Dennis Olivetti, Timothé Picavet, Gustav Schmid

One of the central models in distributed computing is Linial's LOCAL model [SIAM J. Comp. 1992]. Over time, researchers have studied distributed graph problems in the LOCAL model under slightly different assumptions, such as whether nodes know the exact network size $n$, only a polynomial upper bound on $n$, or nothing at all. We ask whether these differences are merely technical or fundamentally affect the theory of Locally Checkable Labelings (LCLs), one of the most studied problem classes. LCLs are graph problems whose valid solutions can be characterized by a finite set of allowed constant-radius neighborhoods. Since their introduction by Naor and Stockmeyer [FOCS 1995], they have become central in distributed computing, and the last decade has seen major progress in understanding their complexity. For example, Chang, Kopelowitz, and Pettie [FOCS 2016] showed that the randomized complexity of any LCL on $n$-node graphs is at least its deterministic complexity on $\sqrt{\log n}$-node graphs. Later, Chang and Pettie [FOCS 2017] showed that any randomized $n^{o(1)}$-round algorithm for LCLs on bounded-degree trees can be turned into a deterministic $O(\log n)$-round algorithm. Then, Balliu et al. [STOC 2018] showed that such automatic speedups are impossible for general bounded-degree graphs. However, these results fundamentally rely on nodes knowing $n$. How much does this assumption affect the theory of LCLs? Our work shows that if nodes are oblivious to $n$, or know only a polynomial upper bound on it, then even on trees, the theory of LCLs changes significantly. While the fundamental classification of problems remains the same, we show the landscape becomes much more complex: for example, for LCLs, randomness helps in more cases; some problems have very unnatural complexities; and some have a lower bound that depends on which definition of $Ω$ we use!

Authors: Alexandru Gheorghiu, Dale Jacobs, Saeed Mehraban, Arsalan Motamedi

We study the computational complexity of quantum state isomorphism problems under group actions: given two quantum circuits that prepare pure or mixed states, decide whether the two states are related by a group action. This can be seen as a quantum state version of the Hidden Shift Problem, in much the same way that the State Hidden Subgroup Problem is a quantum version of the ordinary Hidden Subgroup Problem. We prove several results for this computational problem: - For the pure-state version, we show that the problem is BQP-hard for all nontrivial groups, and contained in QCMA $\cap$ QCSZK. We further obtain refined results for specific groups of interest: for abelian groups we show that the problem reduces to the state hidden subgroup problem over the generalized dihedral group; for the Clifford group, the problem is at least as hard as Graph Isomorphism under polynomial-time reductions; for the Pauli group it is BQP-complete. - For the mixed-state version, for nontrivial, finite and efficiently representable groups, the problem is QSZK-complete. - We also study a variant of this problem over an infinite group, in particular, the bosonic linear optical unitaries. We show that in the setting where the classical description of the quantum state is given in a suitable wave function representation known as the stellar representation, the problem is at least as hard as Graph Isomorphism, and is contained in NP $\cap$ SZK. Prior to our work, state isomorphism problems had only been studied for the symmetric group [LG17]. As a consequence of our results, we resolve an open question posed in [HEC25] about the existence of a quantum algorithm for the abelian state hidden subgroup problem on mixed states. We show that this problem is QSZK-hard in the worst case, thereby ruling out an efficient quantum algorithm unless QSZK = BQP.

Authors: Dongzhe Zheng, Tao Zhong, Christine Allen-Blanchette

In this paper, we study solution operators of physical field equations on geometric meshes from a function-space perspective. We reveal that Hodge orthogonality fundamentally resolves spectral interference by isolating unlearnable topological degrees of freedom from learnable geometric dynamics, enabling an additive approximation confined to structure-preserving subspaces. Building on Hodge theory and operator splitting, we derive a principled operator-level decomposition. The result is a Hybrid Eulerian-Lagrangian architecture with an algebraic-level inductive bias we call Hodge Spectral Duality (HSD). In our framework, we use discrete differential forms to capture topology-dominated components and an orthogonal auxiliary ambient space to represent complex local dynamics. Our method achieves superior accuracy and efficiency on geometric graphs with enhanced fidelity to physical invariants. Our code is available at https://github.com/ContinuumCoder/Hodge-Spectral-Duality

Authors: Martino Bernasconi, Matteo Castiglioni, Andrea Celli, Alexandros Hollender

We study the query complexity of min-max optimization of a nonconvex-nonconcave function $f$ over $[0,1]^d \times [0,1]^d$. We show that, given oracle access to $f$ and to its gradient $\nabla f$, any algorithm that finds an $\varepsilon$-approximate stationary point must make a number of queries that is exponential in $1/\varepsilon$ or $d$.

Authors: Farzan Byramji, Daniel M. Kane, Jackson Morris, Anthony Ostuni

We construct an explicit distribution $\mathbf{D}$ over $\{0,1\}^N$ that exhibits an essentially optimal separation between adaptive and non-adaptive cell-probe sampling. The distribution can be sampled exactly when each output bit is allowed two adaptive probes to an arbitrarily long sequence of independent uniform symbols from $[N]$. In contrast, any non-adaptive sampler requires $\widetildeΩ(N)$ non-adaptive cell probes to generate a distribution with total variation distance less than $1-o(1)$ from $\mathbf{D}$. This provides a $2$-vs-$\widetildeΩ(N)$ separation for sampling with adaptive versus non-adaptive cell probes, improving upon the $2$-vs-$\widetildeΩ(\log N)$ separation of Yu and Zhan (ITCS '24) and the $(\log N)^{O(1)}$-vs-$N^{Ω(1)}$ separation of Alekseev, Göös, Myasnikov, Riazanov, and Sokolov (STOC '26).

Authors: Joshua M. Courtney

Multivariate quantum signal processing (M-QSP) has recently been shown to be applicable for non-Hermitian Hamiltonian simulation, opening several problems regarding the optimization landscape, angle-finding, and constant-factor analysis. We resolve several of these problems here. We find the anti-Hermitian query complexity $d_I = Θ(\betaI T + \log(1/\varepsilon)/\log\log(1/\varepsilon))$ to be tight, established via Chebyshev coefficient bounds, modified Bessel function asymptotics, and Lambert~$W$ inversion. Fast-forwarding to $d_I = \mathcal{O}(\sqrt{\betaI T})$ is impossible in the bivariate polynomial model, though a linear state-dependent improvement to $d_I = \mathcal{O} β_{\mathrm{eff}} T + \log(1/\varepsilon)/\log\log(1/\varepsilon))$ is achievable. The optimization landscape of M-QSP admits spurious local minima, but a warm-start basin guarantee ensures the two-stage algorithm converges. CRC-exploiting block peeling reduces angle-finding from $\mathcal{O}(d^3)$ to $\mathcal{O}(d^2)$ classical operations, and optimized error allocation yields a leading constant of approximately~$2$ relative to the information-theoretic lower bound. A constant-ratio condition extends to non-identical signal operators, enabling time-dependent non-Hermitian simulation with query complexity $\mathcal{O}(\int_0^T(\alphaR(s) + \betaI(s))\,ds + \log(1/\varepsilon)/\log\log(1/\varepsilon))$. Block-encoding overhead $e^{-2\betaI T}$ holds across all function classes within the walk-operator oracle model, and dilational methods (Schrödingerization) achieve the walk-operator barrier. A precisely characterized direct-access construction achieves the intrinsic barrier $e^{-2ωT}$ (with $ω< \betaI$ for non-commuting Hamiltonians) on a restricted domain, though extension to the full bitorus remains open.

Authors: Steve Hanneke, Anay Mehrotra, Grigoris Velegkas, Manolis Zampetakis

Valiant's 1984 paper is widely credited with introducing the PAC learning model, but it, in fact, introduced a different model: unlike PAC learning, the learner receives only positives, may issue membership queries, and must output a hypothesis with no false positives. Prior work characterized variants, including the case without queries. We revisit Valiant's original model and ask: *Which classes are learnable in it?* For every finite domain, including Valiant's Boolean-hypercube setting, we show that a class is learnable if and only if every realizable positive sample can be certified by a poly-size adaptive query-compression scheme. This is a new variant of sample compression where the learner certifies samples via a short interaction with the membership oracle. Our characterization shows that learnability in Valiant's model is strictly sandwiched between learnability in the PAC model and the variant of Valiant's model without membership queries. This is one of the rare cases where introducing membership queries changes the set of learnable classes, and not just the sample or computational complexity. Next, we study the natural extension of the model to arbitrary domains. While we do not obtain an exact characterization, our techniques readily generalize and show that the same strict sandwiching persists. Finally, we show that $d$-dimensional halfspaces, which are not learnable without queries, are learnable with queries: we give a $\mathrm{poly}(d) \tilde{O}(1/ε)$ sample and $\mathrm{poly}(d) \mathrm{polylog}(1/ε)$ query algorithm, and prove that at least $Ω(d)$ samples or queries are necessary. To our knowledge, this is the first algorithm for halfspaces in Valiant's model. Together, these results uncover a surprisingly rich theory behind Valiant's original notion of learnability and introduce ideas that may be of independent interest in learning theory.

Authors: Ying Feng, Piotr Indyk, Michael Kapralov, Dmitry Krachun, Boris Prokhorov

Vector quantization via random projection followed by scalar quantization is a fundamental primitive in machine learning, with applications ranging from similarity search to federated learning and KV cache compression. While dense random rotations yield clean theoretical guarantees, they require $Θ(d^2)$ time. The randomized Hadamard transform $HD$ reduces this cost to $O(d \log d)$, but its discrete structure complicates analysis and leads to weaker or purely empirical compression guarantees. In this work, we study a variant of this approach: dithered quantization with a single randomized Hadamard transform. Specifically, the quantizer applies $HD$ to the input vector and subtracts a random scalar offset before quantizing, injecting additional randomness at negligible cost. We prove that this approach is unbiased and provides mean squared error bounds that asymptotically match those achievable with truly random rotation matrices. In particular, we prove that a dithered version of TurboQuant achieves mean squared error $\bigl(π\sqrt{3}/2 + o(1)\bigr) \cdot 4^{-b}$ at $b$ bits per coordinate, where the $o(1)$ term vanishes uniformly over all unit vectors and all dimensions as the number of quantization levels grows.

Authors: Dipan Dey, Telikepalli Kavitha

Our input is a directed graph $G = (V,E)$ on $n$ vertices and $m$ edges with a designated root vertex $r$ and a function $cost: E \rightarrow \mathbb{R}_{\geq 0}$. The problem is to maintain a min-cost arborescence in $G$ in the presence of edge faults (a single fault at a time). Edge faults are transient and once the faulty edge is repaired, the original min-cost arborescence $\mathcal{T}$ is restored. Whenever an edge fault happens, we need to update $\mathcal{T}$ to a min-cost arborescence in $G-f$, where $f$ is the faulty edge. Since computing a min-cost arborescence in $G - f$ takes $O(m + n\log n)$ time, we seek to construct a sparse subgraph $H$ in a preprocessing step such that in the event of any edge $f$ failing, it suffices to compute a min-cost arborescence in $H - f$ in order to find a low-cost arborescence in $G - f$. In the unweighted setting, this is the fault-tolerant subgraph problem for single-source {\em reachability}. Baswana, Choudhary, and Roditty (SICOMP, 2018) showed a $k$-fault tolerant reachability subgraph of size $O(2^kn)$, where $k$ is the number of edge faults. We show a simple polynomial-time algorithm to construct a subgraph $H$ of size $O(n^{3/2})$ such that, for any $f \in E$, a min-cost arborescence in $H-f$ is a 2-approximation of a min-cost arborescence in $G-f$. Thus whenever an edge fault happens, we can find a 2-approximate min-cost arborescence in $G-f$ in $O(n^{3/2})$ time. Our second problem is in the matroid setting. The input is a matroid $M = (E, {\cal I})$ with a function $cost: E \rightarrow \mathbb{R}$. The problem is to compute a sparse $S \subseteq E$ (called a $k$-fault tolerant preserver) such that for any $F \subseteq E$ with $|F| \le k$, the matroid $M|(S\setminus F)$ contains a min-cost basis of $M|(E\setminus F)$. We show a tight bound of $k.rank(E)$ on the size of a $k$-fault tolerant preserver.

Authors: Christian Møller Mikkelstrup, Anders Bjorholm Dahl, Philip Bille, Vedrana Andersen Dahl, Inge Li Gørtz

Computing a minimum $s$-$t$ cut in a graph is a solution to a wide range of computer vision problems, and is often done using the Boykov-Kolmogorov (BK) algorithm. In this paper, we revisit the BK algorithm from both a theoretical and practical point of view. We improve the analysis of the time complexity of the BK algorithm to $O(mn|C|)$ and propose a new algorithm, the fast and compact BK (fcBK) algorithm, with a time complexity of $O(m|C|)$, where $m$, $n$, and $|C|$ are the number of edges, number of vertices, and the capacity of the cut, respectively. We additionally propose a compact graph representation that allows our implementation to find a minimum $s$-$t$ cut in a graph with upwards of $10^9$ vertices and $10^{10}$ edges on a machine with 128 GB of memory. We find our implementation of the BK algorithm to be the fastest available implementation of the BK algorithm when evaluating on a comprehensive set of benchmark datasets, highlighting the importance of memory-efficient implementations. We make our implementations publicly available for further research and implementation development within minimum $s$-$t$ cut algorithms.

Authors: Asaf Lev-Ran, Pavel Arkhipov, Vladimir Kolmogorov

We consider the MAP-MRF inference task, that is, minimizing a function of discrete variables represented as a sum of unary and pairwise terms. A prominent approach for tackling this NP-hard problem in practice is to solve its natural LP relaxation and then iteratively tighten the relaxation by adding clusters. Based on some theoretical observations, we propose a new technique for identifying such clusters. It works by running the Singleton Arc Consistency algorithm in a certain CSP instance. Experimental results indicate that the new tightening technique outperforms the previous approach by [Sontag et al. UAI 2012] that searches for frustrated cycles. Our code will be made available at https://github.com/vnk-ist/MAP-MRF/.

Authors: Samuel German

A vertex-coloring of a connected graph $G$ is a strong conflict-free vertex-connection coloring if every two distinct vertices are joined by a shortest path on which some color appears exactly once. The minimum number of colors in such a coloring is the strong conflict-free vertex-connection number $\operatorname{svcfc}(G)$. We study this problem under the parameter twin cover. Let $X$ be a twin cover of $G$ of size $t$, and let $k$ be the target number of colors. In our first result, given $(G,k)$ together with a twin cover $X$, we reduce in polynomial time to an equivalent annotated instance on at most $\max\{2,t+(t+1)k2^{t+k-1}\}$ vertices. Hence the annotated version of Strong CFVC Number, in which a twin cover is supplied as part of the input, is fixed-parameter tractable parameterized by $t+k$. Using this bound, we then obtain a kernel parameterized by $\operatorname{tc}(G)+k$; in particular, for every fixed $k$, the problem is fixed-parameter tractable parameterized by the twin-cover number alone. In our second result, we prove every connected graph $G$ with twin cover $X$ of size $t$ satisfies $χ(G)\le \operatorname{svcfc}(G)\le χ(G)+t$. More generally, if $Y\subseteq X$ intersects every shortest path of length at least $3$, then $\operatorname{svcfc}(G)\le χ(G)+|Y|$. We also derive an exact expression for the chromatic number on graphs of bounded twin-cover number: for every proper coloring $\varphi$ of $G[X]$, the minimum number of colors needed to extend $\varphi$ to all of $G$ is $K_\varphi=\max_{S\subseteq X}(|\varphi(S)|+m(S))$, and hence $χ(G)=\min_{\varphi\text{ proper on }G[X]} K_\varphi$. Our results provide the first evidence that twin cover is a useful parameter for strong conflict-free vertex-connection and show that, once a twin cover is fixed, the remaining difficulty is concentrated in a bounded additive gap above the chromatic number.

Authors: Peter Davies-Peck

The classic lower bound of Kuhn, Moscibroda and Wattenhofer [JACM 2016] states that approximate maximum matching and approximate vertex cover (among other problems) in the LOCAL model require $Ω(\min\{\sqrt{\frac{\log n}{\log\log n}}, \frac{\log Δ}{\log\log Δ}\})$ rounds, for any polylogarithmic or smaller approximation ratio. As a function of $Δ$, this complexity was subsequently matched for constant-approximate weighted vertex cover [Bar-Yehuda, Censor-Hillel and Schwartzman, JACM 2017] and constant-approximate maximum matching [Bar-Yehuda, Censor-Hillel, Ghaffari and Schwartzman, PODC 2017]. One might expect, therefore, that the true complexity should be $Θ(\frac{\log Δ}{\log\log Δ})$, and the $n$-dependent term in the lower bound is just an artefact of the proof method. We show that this is not the case, and a term dependent on $n$ is in fact required. Specifically, we show randomized algorithms for $2+\varepsilon$-approximate maximum matching and approximate (weighted) minimum vertex cover taking $O(\frac{\log n}{\log^2 \log n})$ rounds. Our algorithms are based on a novel graph decomposition result generalizing the method of Miller, Peng and Xu [SPAA 2013], which we use to reduce the `effective' degree of high-degree graphs. We expect that this decomposition may be of further use for other problems.

Authors: Sebastian Schmidt, Juha Harviainen, Corentin Moumard, Aleksandr Politov, Francisco Sena, Alexandru I. Tomescu

Bidirected graphs are a common generalisation of directed graphs where arcs can also be incoming to both their incident nodes, or outgoing from both their incident nodes. Such arcs allow a walk to change direction. Some algorithms can easily be adapted from directed graphs to bidirected graphs, such as shortest path algorithms. These adaptions are already used in practice, and implicitly use the graph doubling technique to apply an algorithm for directed graphs to bidirected graphs. In other cases, the applicability of graph doubling is not that obvious. For example, superbubbles and their generalisation to bidirected graphs ultrabubbles. Ultrabubbles are a common structure in bidirected biological graphs which carries biological meaning, but also functions as a nested clustering method, since an ultrabubble is separated by only two nodes from the rest of the graph. There is an existing method that enumerates a structure similar to ultrabubbles by enumerating (weak) superbubbles in the doubled graph. However, the literature does not make any direct connection between superbubbles and ultrabubbles except that a superbubble is an ultrabubble in a directed graph. Only a partial result connecting superbubbles and ultrabubbles exists by Harviainen et al. (2026). Graph doubling on the other hand maintains connectivity, and allows to draw a direct connection between ultrabubbles and weak superbubbles. This results in the first linear-time reduction-based algorithm for computing ultrabubbles on any bidirected graph. Together with the fact that graph doubling is already used implicitly in simple cases, our result motivates that graph doubling is a powerful yet simple technique to apply algorithms for directed graphs to bidirected graphs.

Authors: Dinesh Kumar Baghel, Alex Ravsky, Erel Segal-Halevi

Given items of different sizes and a fixed bin capacity, the bin-packing problem is to pack these items into the minimum number of bins such that the sum of the item sizes in each bin does not exceed the capacity. We define a new variant, k-times bin-packing (kBP), in which the goal is to pack the items so that each item appears exactly k times in k different bins. We generalize existing approximation algorithms for bin-packing to solve kBP and analyze their performance ratios. The fair electricity division problem motivates the study of kBP. The goal is to allocate the available supply among households using some fairness criteria, such as the egalitarian principle. We prove that every electricity division problem can be solved by k-times bin-packing for some finite k, which depends only on the number of households. We implement generalizations of the First-Fit and First-Fit Decreasing bin-packing algorithms to solve kBP and apply them to real electricity demand data. We show that our generalizations outperform existing heuristic solutions to the same problem in terms of the egalitarian allocation of connection time. We study another variant of the egalitarian allocation problem, in which the goal is to maximize the minimum number of watts allocated to a household. For this variant, we prove an impossibility result: there does not exist such a k that depends only on the number of agents. This impossibility result motivates us to develop four different heuristic algorithms to solve the egalitarian allocation of watts problem. We evaluate the heuristics by summing the minimum watts allocated to any household in each hour, yielding a fairness metric that reflects the lowest watt allocation across all hours. A higher total minimum of watts indicates a more equitable distribution. Thus, we establish new benchmarks for fair allocation of watts.

Authors: Nathan Klein, Neil Olver, Zi Song Yeoh

The strong thin tree conjecture states that every $k$-edge-connected graph $G$ contains an $O(1/k)$-thin spanning tree, meaning a spanning tree which contains at most an $O(1/k)$ fraction of the edges across each cut in $G$. This conjecture is still open despite significant effort; the best current result by Anari and Oveis Gharan shows the existence of an $O(\text{polyloglog}(n)/k)$-thin tree. In this work, we demonstrate that the conjecture is true if one only requires thinness for the set of $η$-near minimum cuts of the graph for $η= 1/40$, in other words, for the set of cuts with fewer than $(1+1/40)k$ edges. Our approach constructs such a tree in polynomial time. To show this, we utilize the structure of near minimum cuts, and in particular the polygon representation of Benczúr and Goemans, to reduce to the previously solved problem of finding a spanning tree that is $O(1/k)$-thin for all sets in a laminar family.

September 7-11, 2026 Cambridge, United Kingdom https://www.newton.ac.uk/event/lfcw01/ Registration deadline: July 19, 2026 We are pleased to announce the workshop “Frontiers in Complexity Lower Bounds”, which will take place at the Isaac Newton Institute for Mathematical Sciences in Cambridge (UK) from 7 to 11 September 2026. Many fundamental questions in complexity theory, such as the P … Continue reading Workshop “Frontiers in Complexity Lower Bounds”

By shacharlovett

Over the years, I’ve written two op-eds for The New York Times about quantum computing, at the NYT editors’ invitation:

• Quantum Computing Promises New Insights, Not Supermachines (2011)
• Why Google’s Quantum Supremacy Milestone Matters (2019)

I’ve also visited the NYT office and helped NYT reporters with numerous stories about quantum computing and beyond. In the wake of Cade Metz’s infamous NYT hatchet job against Scott Alexander and the rationalist community, I resolved no longer to talk to Metz, but it never even occurred to me to extend that to a broader ban against the NYT itself. After all, it’s the friggin’ NYT!

This week, however, Nicholas Kristof—a man who I praised fulsomely in a blog post 20 years ago, for his coverage of the genocide in Darfur—has used the NYT to broadcast the oldest, crudest form of antisemitic libel, accusing the Jews of garishly preposterous crimes (poisoning wells? baking blood into matzo? in this case, training dogs to rape prisoners). As countless others have since pointed out, the sole source for this ludicrous accusation was a Hamas-linked organization called “Euro-Med” that praised the October 7 “martyrs.” Kristof’s piece came out the same day an Israeli human rights organization released a major report meticulously documenting Hamas’s mass rapes on October 7–something that did happen—and was apparently designed to neutralize the impact of that report.

While the debunking of Kristof came fast, it wasn’t fast enough: now that antisemitic blood libel (dog libel?) has the Gray Lady’s imprimatur, I expect it to ignite violence against Jews all over the world, and I expect my kids to be less safe. So I hereby announce:

I, Scott Aaronson, member of the National Academy of Sciences, will no longer cooperate with anyone from the NYT on anything—neither quantum computing stories nor anything else—until the NYT, at minimum, formally retracts its dog-rape claim and fires Kristof.

To my friends doing good science and technology writing for NYT: I’m sorry, I hope you understand, and please fight for what’s right within the organization.

I say “bare minimum” because I hope this scandal causes a major reckoning for the NYT as an institution. I don’t know if any libel or other laws were broken, but if they were, I hope Kristof personally and the NYT as a whole get sued for whatever they’re worth.

---

Luckily, others have already said most of what I would’ve wanted to. Here’s Eli Lake, in a passage that part of me wishes I’d never read:

Let’s start with what is known about the biology of male dogs. Their penises are small and thin. They become erect only when they smell the pheromones of a female dog in heat. Brandon McMillan, the three-time Emmy-winning host of CBS’s Lucky Dog, who has spent 25 years training animals, told me he had never heard of a dog who was trained to rape a human being and doubted this was possible.

“When a female is in heat, the pheromones released carry it to the male canine,” McMillan said. “That’s how they reproduce and the miracle happens. I don’t see how you would train a dog to do that. The dog has to get turned on, for lack of a better word.”

On the NYT’s “vetting process” (or lack thereof), here’s comedian Jeff Maurer:

Imagine that you’re a fact checker or editor at The New York Times. You know damn well that you’ve nabbed a lifeboat while most of your field is being picked apart by hagfish at the bottom of the North Atlantic. If you’re a lowly fact-checker, you might be in your 20s and desperate to rationalize your choice to enter a field that might as well be buggy repair. I‘ve had similar gigs and been in similar environments; I feel like I can get in these peoples’ heads. And that’s why I’m convinced that the dog claim should have made things easier for the fact checkers.

One day, Nicholas Kristof — the award winning columnist who has been with the Times since it was Farmer Sulzberger’s New Amsterdam Seed Catalogue — comes to you, the lowly fact checker. He’s working on a big piece, and it’s scorching hot: It’s sexual assault plus Israel/Palestine, which is like detonating a nuclear weapon inside a volcano. You know it will be very tough to say “I don’t find this source credible” or “we can’t run that without more substantiation”. And that’s true even before we consider that you probably run in a social circle in which Israel is thought of as a substantially-more-evil version of The Borg Collective from Star Trek (which is why they have those vaporizing weapons).

But you do notice that Kristof’s sources are thin. In fact, he only has two named sources. The first source tells him a story that has substantially changed since he told it to The Washington Post two years ago. The second source celebrated the October 7 attacks and previously made torture claims that he later recanted. Much of the testimony contains no detail that could be corroborated. The organizations and journalists that Kristof relies on are highly dubious and engaged in the mutual citation circle jerk that is typically the way that lies enter the zeitgeist. There may be some truth in there somewhere, but Kristof’s methods are — what’s the journalistic term? — total ass. But you’re afraid that if you question things, you’ll be thought of as pro-rape, or worse yet: pro-Israel.

And in that context, the dog story is a godsend. It frees you from addressing the broader problems with the article, because you can pick the battle of focusing on the most-outrageous, least-believable claim. The article doesn’t need the dog rape detail — it’s already full of shocking-if-true claims. Striking that one section lets you maintain a sheen of professionalism, and when the thing explodes, you also get to say “You should have seen what he wanted to print”.

The fact that that didn’t happen tells me that the level of scrutiny for claims about Israel at the Times is absolute zero. Even if some Israeli somewhere did manage to turn his dog into a sex criminal, there is not nearly enough evidence for that claim to run in the Times. It’s an unsubstantiated rumor, and a vile one. I think that a person who believes that without compelling evidence is showing evidence of a damaged mind. And I think that a newspaper that publishes that without compelling evidence is showing evidence of a damaged vetting process, or perhaps a vetting process that — when it comes to claims about Israel — doesn’t exist at all.

Last but not least, here’s Haviv Rettig Gur, with righteous anger entirely appropriate to the situation, yet also a determination to expose and fight prisoner abuse, which is a genuine problem in Israel as it is in pretty much every other country on earth:

No, dogs aren’t being trained to systematically rape prisoners, you nattering halfwits. And no, Hamas propaganda operatives are not reliable sources on the question of Israeli crimes. The vast, vast majority of soldiers are honorable men who walked into fire so our families may live. The whole world may turn on them; I will stand with them, grateful for their sacrifice. And Kristof, a willing purveyor of propaganda happily feigning that he can’t see the water and thrilling to a moral crusade engineered by would-be genocidaires he pretends not to understand — is no messenger of moral reckoning.

But friends, so fucking what. Let the narcissistic guttersnipes strut their moral emotions before the world, let the UN publish endless reports that don’t hold up to basic scrutiny, let the NGOs dream their rabid, sick dreams that no journalist ever fact-checks — yes, they’re lying. But so fucking what.

We still, for ourselves — because fuck them — must see that it isn’t all fake. The problem [of abuse in Israeli prisons] is real. It’s far smaller than they claim, but real nonetheless. And when discipline and morality break down, it can only get worse. We either crack down now or we watch it fester and grow.

And our own Ben Gvirs are stubbornly refusing to fix what is actually broken, the real thing in the real world.

And so we are caught in a strange sort of vise, the same vise we find ourselves in with the genocide lie: A vast propaganda machine that seeks to destroy us — countless activists too high on their own self-regard to see the irony of raging against a ‘genocide’ while calling for the erasure of a people — all while our own incompetent, venal, self-absorbed political class insist in their mindless chatter on confirming every claim of our enemies for sheer, bald egomania.

I’m sick of it all. I know you’re all sick of it too.

And that, in a nutshell, is what I think about this.

Just because they’re lying, just because a vast perfidious campaign has overwhelmed global elites in a bid to clear the way for our removal, just because they’re still, after two millennia, building their visions of redemption on The Evil Jew — doesn’t mean there isn’t also, separately, a problem on the ground.

So what do we do now? Simple. We see it, we acknowledge it’s happening, we bring our rage to our inept leaders until they bend to our will and act to stop the breakdown…

And we soldier on.

We soldier on because the enemy really is coming to murder us. Because Hamas must still go if Gaza is ever to rise to a new day. Because Hezbollah will yet destroy Lebanon on the altar of destroying us. Because the ayatollahs built their whole damn religion on the extermination of our children.

We fix the broken things within us as if the pogromists and their simpering Kristofs don’t exist. We owe no answers to the propagandists who seek to clear a path to our deaths. But we do owe answers to ourselves.

Let the screaming mob rage and churn like so much sea-foam. Despite that raging mob, despite the enemy who still seeks our destruction, and yes, despite feckless incompetents like Ben Gvir, our minister of prisons, who claim to lead us — we remain the strongest, freest Jews who ever lived, more capable and committed than our self-destructive enemies ever imagined. And the task is still before us, yet to be completed, the sacred duty given to our generation to ensure our children don’t have to face the genocidaires who now surround us. We do not waver, we do not stumble. We soldier on.

Because fuck them all.

---

Inspired by Haviv’s energy, I’m leaving the comments on this post turned off. Why? Because outside the dog-rape libel, I was having a pretty good week! On Monday I visited Quantinuum in Colorado and got a personal tour of Helios 1, considered the all-around most powerful quantum computer on earth at the moment. Now I’m visiting Stony Brook University, where I’ll give two talks and meet lots of interesting people. Then I’m headed to NYC for the annual meeting of Quanta magazine’s advisory board.

Right now, then, I have neither the time nor the inclination to battle the infinite army of brain-eaten zombies who arrive for every single post touching on Israel. My life will be better this way.

By Scott

We construct an explicit distribution $\mathbf{D}$ over $\{0,1\}^N$ that exhibits an essentially optimal separation between adaptive and non-adaptive cell-probe sampling. The distribution can be sampled exactly when each output bit is allowed two adaptive probes to an arbitrarily long sequence of independent uniform symbols from $[N]$. In contrast, any non-adaptive sampler requires $\tilde{\Omega}(N)$ non-adaptive cell probes to generate a distribution with total variation distance less than $1-o(1)$ from $\mathbf{D}$. This provides a $2$-vs-$\tilde{\Omega}(N)$ separation for sampling with adaptive versus non-adaptive cell probes, improving upon the $2$-vs-$\tilde{\Omega}(\log N)$ separation of Yu and Zhan (ITCS '24) and the $(\log N)^{O(1)}$-vs-$N^{\Omega(1)}$ separation of Alekseev, Göös, Myasnikov, Riazanov, and Sokolov (STOC '26).

Authors: Matthäus Micun, Christoph Berkholz

Since their introduction by Atserias, Kolaitis, and Vardi in 2004, proof systems where each line is represented by an ordered binary decision diagram (OBDD) have been intensively studied as they allow to compactly represent Boolean functions. We extend this line of work by considering representation formats that can be even more succinct than OBDDs and have gained a lot of attention in the area of knowledge compilation: sentential decision diagrams (SDDs) and deterministic structured DNNF circuits (d-SDNNFs). We show that both variants can provide strictly smaller refutations of unsatisfiable CNFs than their OBDD counterparts. Furthermore, we investigate the relative strength of these systems depending on which of the three fundamental derivation rules join, reordering, and weakening are allowed. Here we obtain several separations and identify interesting open problems. To streamline our proofs we establish a sat-to-unsat lifting theorem that might be of independent interest: it turns satisfiable CNFs that are hard to represent by SDDs and d-SDNNFs into unsatisfiable CNFs that are hard to refute in the corresponding proof system.

Authors: Leif Eriksson, Victor Lagerkvist, Sebastian Ordyniak, George Osipov, Fahad Panolan, Mateusz Rychlicki

Determining the validity of a quantified Boolean formula (QBF) is a PSPACE-complete problem with rich expressive power. Despite interest in efficient solvers, there is, compared to problems in NP, a lack of positive theoretical results, and in the parameterized complexity setting one often has to restrict the quantifier prefix (e.g., bounding alternations) to obtain fixed parameter tractability (FPT). We propose a new parameter: the number of variables in clauses that has to be removed before reaching a tractable class (a clause covering (CC) backdoor). We are then interested in solving QBF in FPT time given a CC-backdoor of size $k$. We consider the three classical, tractable cases of QBF as base classes: Horn, 2-CNF, and linear equations. We establish W[1]-hardness for Horn but prove FPT for the others, and prove that in a precise, algebraic sense, we are only missing one important case for a full dichotomy. Our algorithms are non-trivial and depend on propagation, and Gaussian elimination, respectively, and are comparably unexplored for QBF.

Authors: Venkatesan Guruswami, Xuandi Ren, Shaoxuan Tang

Given a linear subspace of $n \times n$ matrices over $\mathbb F_{2^r}$ that is promised to contain a matrix of rank $1$, we prove that it is hard to find a matrix of rank $n^{o(1/\log \log n)}$, assuming NP doesn't have sub-exponential algorithms. In addition to being a basic problem, the hardness of this problem, even for the exact version, drove recent PCP-free inapproximability results for minimum distance and shortest vector problems concerning codes and lattices. The proof combines the concept of superposition soundness introduced by Khot and Saket with moment matrices. To produce a rank-gap of $1$ vs. $k$, the reduction runs in time $n^{O(\log k)}$. We also give another moment-matrix-based construction which runs in time $n^{O(k)}$ but works for any finite field $\mathbb F_q$.

Authors: Tian Bai, Yixin Cao, Mingyu Xiao

The feedback set problems are about removing the minimum number of vertices or edges from a graph to break all its cycles. Much effort has gone into understanding their complexity on planar graphs as well as on graphs of bounded degree. We obtain a complete complexity classification for these problems on bounded-degree digraphs, including the planar case. In particular, we show that both problems are $\NP$-complete on digraphs of maximum degree three, while on planar digraphs the feedback vertex set problem is polynomial-time solvable when each vertex has either indegree at most one or outdegree at most one, and $\NP$-complete otherwise. We also give tight degree bounds for the connected feedback vertex set problem on undirected graphs, both planar and non-planar. We close the paper with a historical account of results for feedback vertex set on undirected graphs of bounded degree.

Authors: Todor Antić, Vít Jelínek, Jan Kratochvíl, Peter Stumpf

An \emph{outer-string representation} of a graph $G$ is an intersection representation of $G$ where vertices are represented by curves (strings) inside the unit disk and each curve has exactly one endpoint on the boundary of the unit disk (the anchor of the curve). Additionally, if each two curves are allowed to cross at most once, we call this an \emph{outer-$1$-string representation} of $G$. If we impose a cyclic ordering on the vertices of $G$ and require the cyclic order of the anchors to respect this cyclic order, such a representation is called a \emph{constrained outer-string representation}. In this paper, we present two results about graphs admitting outer-string representations. Firstly, we show that for a bipartite graph $G$ (and, more generally, for any $\{C_3,C_5\}$-free graph $G$) with a given cyclic order of vertices, we can decide in polynomial time whether $G$ admits a constrained outer-string representation. Our algorithm follows from a characterization by a single forbidden configuration, similar to that of Biedl et al. [GD 2024] for chordal graphs. Secondly, we answer an open question from the same authors and show that determining whether a given graph admits an outer-1-string representation is NP-hard. More generally, we show that it is NP-hard to determine if a given graph $G$ admits an outer-$k$-string representation for any fixed $k\ge1$.

Authors: Joshua M. Courtney

We achieve query-optimal quantum simulations of non-Hermitian Hamiltonians $H_{\mathrm{eff}} = H_R + iH_I$, where $H_R$ is Hermitian and $H_I \succeq 0$, using a bivariate extension of quantum signal processing (QSP) with non-commuting signal operators. The algorithm encodes the interaction-picture Dyson series as a polynomial on the bitorus, implemented through a structured multivariable QSP (M-QSP) circuit. A constant-ratio condition guarantees scalar angle-finding for M-QSP circuits with arbitrary non-commuting signal operators. A degree-preserving sum-of-squares spectral factorization permits scalar complementary polynomials in two variables. Angles are deterministically calculated in a classical precomputation step, running in $\mathcal{O}(d_R \cdot d_I)$ classical operations. Operator norms $α_R\,,β_I$ contribute additively with query complexity $\mathcal{O}((α_R + β_I)T + \log(1/\varepsilon)/\log\log(1/\varepsilon))$ matching an information-theoretic lower bound in the separate-oracle model, where $H_R$ and $H_I$ are accessed through independent block encodings. The postselection success probability is $e^{-2β_I T}\|e^{-iH_{\mathrm{eff}}T}|ψ_0\rangle\|^2\cdot (1 - \mathcal{O}(\varepsilon))$, decomposing into a state-dependent factor $\|e^{-iH_{\mathrm{eff}}T}|ψ_0\rangle\|^2$ from the intrinsic barrier and an $e^{-2β_I T}$ overhead from polynomial block-encoding.

Authors: Marco Bazzani, Sanjoy Dasgupta

The $k$-d tree is one of the oldest and most widely used data structures for nearest neighbor search. It partitions Euclidean space into axis-aligned rectangular cells. There are two standard ways to find the nearest neighbor to a query in a $k$-d tree. Defeatist search returns the closest data point in the query's cell, while comprehensive search also searches other cells as needed to guarantee it finds the nearest neighbor. Both strategies are commonly believed to perform poorly in high dimensions, but there have been few theoretical results explaining this. We prove non-asymptotic bounds on the runtime of comprehensive search and the accuracy of defeatist search. Under mild distributional assumptions, when the dimension $d$ is at least polylogarithmic in the number of data points, defeatist search is no more likely to return the nearest neighbor than random guessing, and comprehensive search visits every cell with high probability. We also show that on uniform data, with high probability, comprehensive search visits at most $2^{\mathcal{O}(d)}$ cells when each cell contains at least logarithmically many data points, and defeatist search returns the nearest neighbor when each cell additionally contains at least $2^{\mathcal{O}(d \log d)}$ data points. Finally, for arbitrary absolutely continuous distributions, we upper bound the expected distance between the query and the point returned by defeatist search.

Authors: Linghai Liu, Sinho Chewi

We propose an algorithm to sample from composite log-concave distributions over $\mathbb{R}^d$, i.e., densities of the form $π\propto e^{-f-g}$, assuming access to gradient evaluations of $f$ and a restricted Gaussian oracle (RGO) for $g$. The latter requirement means that we can easily sample from the density $\text{RGO}_{g,h,y}(x) \propto \exp(-g(x) -\frac{1}{2h}||y-x||^2)$, which is the sampling analogue of the proximal operator for $g$. If $f + g$ is $α$-strongly convex and $f$ is $β$-smooth, our sampler achieves $\varepsilon$ error in total variation distance in $\widetilde{\mathcal O}(κ\sqrt d \log^4(1/\varepsilon))$ iterations where $κ:= β/α$, which matches prior state-of-the-art results for the case $g=0$. We further extend our results to cases where (1) $π$ is non-log-concave but satisfies a Poincaré or log-Sobolev inequality, and (2) $f$ is non-smooth but Lipschitz.

Authors: Samuel German

The Path Avoiding Forbidden Pairs problem (PAFP) asks whether, in a directed graph $G$ with terminals $s,t$ and a set $\mathcal{F}$ of forbidden vertex pairs, there is an $s$-$t$ path that contains at most one endpoint from each forbidden pair. We initiate the study of PAFP through a layer-based width measure. Our first focus is the union digraph $G\cup\mathcal{F}$, obtained by adding to $G$ one arc per forbidden pair, oriented according to a fixed reachability-compatible order. Let the BFS layer $L_d$ be all vertices at directed shortest-path distance $d$ from $s$, where the BFS-width from $s$ is $\max_d |L_d|$. We show if $G\cup\mathcal{F}$ has BFS-width $b$ from $s$ and only $β$ arcs going from a later BFS layer to an earlier one, then PAFP is FPT parameterized by $b+β$. The backward-arc hypothesis is essential: we show PAFP remains NP-complete when the union digraph is a DAG with BFS-width 2. We also show if the input DAG has BFS-width at most $2$ and only $k$ backward input arcs, then PAFP can be decided in $2^k |I|^{O(1)}$ time, with unrestricted forbidden pairs. This width-$2$ result is tight: inspection of a classical reduction shows NP-completeness on input DAGs of BFS-width $3$ with no backward input arcs. Moreover, we study exact-length layers in the input graph, where the $d$-th layer consists of the vertices reachable from $s$ by a directed path of length exactly $d$. For DAGs of exact-length width at most $2$, we show PAFP is polynomial-time decidable by a 2-SAT encoding of fixed-length paths. This bound is tight: the same classical reduction yields NP-completeness on DAGs of exact-length width $3$. Unlike previously known polynomial-time regimes for PAFP, which restrict the forbidden-pair set in order to obtain tractability, our two input-graph tractability results allow unrestricted forbidden pairs and input graphs with exponentially many $s$-$t$ paths.

Authors: Yuqi Pan, Davin Choo, Haichuan Wang, Milind Tambe, Alastair van Heerden, Cheryl Johnson

We study a sequential resource allocation problem motivated by adaptive network recruitment, in which a limited budget of identical resources must be allocated over multiple rounds to individuals with stochastic referral capacity. Successful referrals endogenously generate future decision opportunities while allocating additional resources to an individual exhibits diminishing returns. We first show that the single-round allocation problem admits an exact greedy solution based on marginal survival probabilities. In the multi-round setting, the resulting Bellman recursion is intractable due to the stochastic, high-dimensional evolution of the frontier. To address this, we introduce a population-level surrogate value function that depends only on the remaining budget and frontier size. This surrogate enables an exact dynamic program via truncated probability generating functions, yielding a planning algorithm with polynomial complexity in the total budget. We further analyze robustness under model misspecification, proving a multi-round error bound that decomposes into a tight single-round frontier error and a population-level transition error. Finally, we evaluate our method on real-world inspired recruitment scenarios.

Authors: Flavio Chierichetti, Mirko Giacchini, Ravi Kumar, Erasmo Tani

Locality-sensitive hashing (LSH) has found widespread use as a fundamental primitive, particularly to accelerate nearest neighbor search. An LSH scheme for a similarity function $S:\mathcal{X} \times \mathcal{X} \to [0,1]$ is a distribution over hash functions on $\mathcal{X}$ with the property that the probability of collision of any two elements $x,y\in \mathcal{X}$ is exactly equal to $S(x,y)$. However, not all similarity functions admit exact LSH schemes. The notion of LSH distortion measures how multiplicatively close a similarity function is to having an LSH scheme. In this work, we study the LSH distortion of the Ulam and Cayley similarities, which are popular similarity measures on permutations of $n$ elements. We show that the Ulam similarity admits a sublinear LSH distortion of $O(n / \sqrt{\log n})$; we also prove a lower bound of $Ω(n^{0.12})$ on the best LSH distortion achievable. On the other hand, we show that the LSH distortion of the Cayley similarity is $Θ(n)$.

Authors: Argyrios Deligkas, Michelle Döring, Eduard Eiben, George Skretas, Georg Tennigkeit

We examine the problem of maximizing the reachability of a given source in temporal graphs that are given as the union of k temporal paths, i.e., every given path is a sequence of edges with strictly increasing labels that denote availability in time. This type of temporal graphs represent train networks. We consider shifting operations on the labels of the paths that maintain their temporal continuity. This means that we can move the availability of a temporal edge later or earlier in time, and propagate the shifts to all other affected edges of the path in order to preserve its temporal connectivity. We study the parameterized complexity of the problem with respect to the number of paths k, and the total budget b, where b is the maximum number of shifts we are allowed to perform. Our results reveal that fixed parameter tractability can be achieved (1) when parameterized both by k and b, and (2) when parameterized by k, and b is unconstrained. In almost every other case, e.g., parameterized by a single parameter or parameterized by k, while having a bound on b, we establish intractability lower bounds that are matched by XP algorithms.

Authors: Tuukka Korhonen, Mikkel Thorup

In the vertex connectivity augmentation problem, we are given an undirected $n$-vertex graph $G$, a set of links $L \subseteq \binom{V(G)}{2} \setminus E(G)$, and integers $λ$ and $k$. The task is to insert at most $k$ links from $L$ to $G$ to make $G$ $λ$-vertex-connected. We show that the problem is fixed-parameter tractable (FPT) when parameterized by $λ$ and $k$, by giving an algorithm with running time $2^{O(k \log (k + λ))} n^{O(1)}$. This improves upon a recent result of Carmesin and Ramanujan [SODA'26], who showed that the problem is FPT parameterized by $k$ but only when $λ\le 4$. We also consider the analogous edge connectivity augmentation problem, where the goal is to make $G$ $λ$-edge-connected. We show that the problem is FPT when parameterized by $k$ only, by giving an algorithm with running time $2^{O(k \log k)} n^{O(1)}$. Previously, such results were known only under additional assumptions on the edge connectivity of $G$.

Authors: Ankur Moitra, Andrej Risteski, Dhruv Rohatgi

Inference-time reward alignment asks how to turn a pre-trained diffusion model with base law $p$ into a sampler that favors a reward $r$ while remaining close to $p$. Since there is no canonical distributional distance for this closeness constraint, different choices lead to different "reward-aligned" laws and, just as importantly, different algorithmic problems. We develop a primitive-based approach to reward alignment: rather than assuming arbitrary reward-aligned laws can be sampled, we ask which simple algorithmic primitives suffice to implement alignment for non-trivial reward classes. If closeness is measured in KL distance, the target law is $q(x) \propto p(x) \exp(λ^{-1}r(x))$. For this setting, we show that linear exponential tilts of the form $q(x)\propto p(x)\exp(\langle θ, x \rangle)$ -- which according to recent work [MRR26] can be efficiently sampled from -- are a sufficient primitive for aligning to a very broad class of convex low-dimensional rewards. If closeness is measured in Wasserstein distance, the corresponding primitive is a proximal transport oracle: given $x$, solve $\mbox{argmax}_y \{r(y)- λc(x,y)\}$. This oracle can be efficiently implemented for concave or low-dimensional Lipschitz rewards $r(x)=f(Ax)$. Together, these results illustrate that the choice of distribution distance for alignment affects the computational primitive and the tractable reward class.

A fully-funded PhD position and Masters positions with RA/TA support are available under the supervision of Prof. Prantar Ghosh at Tennessee Tech University starting Fall 2026. Research will focus broadly on graph algorithms. A good mathematical background and prior experience in Theoretical CS is required. Please email CV and a short description of background to the email below.

Website: https://sites.google.com/view/prantarg/home
Email: pghosh@tntech.edu

By shacharlovett

Authors: Susanna F. de Rezende, David Engström, Yassine Ghannane, Duri Andrea Janett, Artur Riazanov

We study the average-case hardness of establishing that a graph does not have a large clique in both proof and communication complexity. We show exponential lower bounds on the length of cutting planes and bounded-depth resolution over parities refutations of the binary encoding of clique formulas on randomly sampled dense graphs. Moreover, we show that the randomized communication complexity of finding a falsified clause in these formulas is polynomial.

Authors: Argyrios Deligkas, John Fearnley, Alexandros Hollender, Themistoklis Melissourgos

We study the problem of computing approximate market equilibria in Fisher markets with separable piecewise-linear concave (SPLC) utility functions. In this setting, the problem was only known to be PPAD-complete for inverse-polynomial approximations. We strengthen this result by showing PPAD-hardness for constant approximations. This means that the problem does not admit a polynomial time approximation scheme (PTAS) unless PPAD$=$P. In fact, we prove that computing any approximation better than $1/11$ is PPAD-complete. As a direct byproduct of our main result, we get the same inapproximability bound for Arrow-Debreu exchange markets with SPLC utility functions.

Authors: Timo Fritsch, Marvin Künnemann, Mirza Redzic, Julian Stieß

Consider the fundamental task of finding independent sets of (constant) size $k$ in a given $n$-node hypergraph. How is the time complexity affected by the sparsity of the input, i.e., the number of hyperedges $m$? Turán's theorem implies that the problem is trivial if $m=O(n^{2-ε})$ for some $ε> 0$. Above that threshold (i.e., if $m=Θ(n^γ)$ for some $γ\ge 2$), we give a perhaps surprising algorithm with running time $O\left(\min\left\{n^{\fracω{3}k} + m^{k/3}, n^k\right\}\right)$ (for $k$ divisible by 3), which is essentially conditionally optimal for all $γ\ge 2$, assuming the $k$-clique and 3-uniform hyperclique hypotheses (here, $ω<2.372$ denotes the matrix multiplication exponent). In fact, we obtain a more detailed time complexity, sensitive to the arity distribution of the hyperedges. To study such phenomena in more generality, we study the time complexity of finding solutions of (constant) size $k$ in sparse instances of Boolean constraint satisfaction problems, where $n$ and $m$ denote the number of variables and constraints. Our results include an essentially full classification of the influence of sparsity for Boolean constraint families of binary arity. Of particular technical interest is a conditionally tight algorithm for the family consisting of the binary NAND and Implication constraints, with a running time of $Θ(m^{ωk/6 \pm c})$. Further, we identify a large class of constraint families $F$ that exhibits a sharp phase transition: there is a threshold $γ_F$ such that the problem is trivial for $m=O(n^{γ_F-ε})$, but requires essentially brute-force running time $Θ(n^{k\pm c})$ for $m=Ω(n^{γ_F})$, assuming the 3-uniform hyperclique hypothesis. Notably, in many cases the combination of constraints display higher time complexity than either constraint alone.

Authors: Christoph Grüne, Tom Janßen

The problem Defensive $δ$-Covering, for some covering range $δ> 0$, is a continuous facility location problem on undirected graphs where all edges have unit length. It is a generalization of Defensive Dominating Set and $δ$-Covering. An attack and defense are sets of points, which are on vertices or on the interior of an edge. A defense counters an attack, if there is a matching of the points in the defense to the points in the attack, such that any matched points have distance at most $δ$, and every point in the attack is matched. The task is, given a graph $G$ and numbers $\ell, k \in \mathbb N$, to find a defense of size at most $\ell$ that counters every possible attack of size at most $k$. We study the complexity of this problem in various different settings. We show that if the attack is restricted to vertices, the problem is $Σ^P_2$-complete for large $δ$, but if the attack may consist of any points on the graph, it is NP-complete. Additionally, we analyze how the complexity changes if the attacks or defenses may be a multiset. If the defense is allowed to be a multiset, the complexity does not change in any case we consider, while if the attack is allowed to be a multiset, the problem often becomes easier. To show containment in the various complexity classes, we introduce a number of discretization arguments, which show that solutions with a regular structure must always exist.

Authors: Yuhan Ye

We study the parameterized complexity of testing approximate first-order stationarity at a prescribed point for continuous piecewise-affine (PA) functions, a basic task in nonsmooth optimization. PA functions form a canonical model for nonsmooth stationarity testing and capture the local polyhedral geometry that appears in ReLU-type training losses. Recent work by Tian and So (SODA 2025) shows that testing approximate stationarity notions for PA functions is computationally intractable in the worst case, and identifies fixed-dimensional tractability as an open direction. We address this direction from the viewpoint of parameterized complexity, with the ambient dimension $d$ as the parameter. In this paper, we give XP algorithms in fixed dimension for the tractable sides, and prove W[1]-hardness for the complementary sides. Moreover, lower bounds under the Exponential Time Hypothesis rule out algorithms running in time $ρ(d)\size^{o(d)}$ for any computable function $ρ$, where $\size$ denotes the total binary encoding length of the stationarity-testing instance. As a further consequence, our results yield the corresponding parameterized complexity picture for testing local minimality of continuous PA functions. We further extend our hardness results to a family of shallow ReLU CNN training losses, with stationarity tested in the trainable weight space. Thus, the same parameterized-complexity picture also appears for simple CNN training losses.

Authors: Binxu Wang, Emma Lucia Byrnes Finn, Bingbin Liu

Generative models trained on finite data face a fundamental tension: their score-matching or next-token objective converges to the empirical training distribution rather than the population distribution we seek to learn. Using rule-valid synthetic tasks, we trace this tension across two training timescales: $τ_{\mathrm{rule}}$, the step at which generations first become rule-valid, and $τ_{\mathrm{mem}}$, the step at which models begin reproducing training samples. Focusing on parity and extending to other binary rules and combinatorial puzzles, we characterize how these two clocks, $τ_{\mathrm{rule}}$ and $τ_{\mathrm{mem}}$, depend on key aspects of the learning setup. Specifically, we show that $τ_{\mathrm{rule}}$ increases with rule complexity and decreases with model capacity, while $τ_{\mathrm{mem}}$ is approximately invariant to the rule and scales nearly linearly with dataset size $N$. We define the \emph{innovation window} as the interval $[τ_{\mathrm{rule}}, τ_{\mathrm{mem}}]$. This window widens with increasing $N$ and narrows with rule complexity, and may vanish entirely when $τ_{\mathrm{rule}} \geq τ_{\mathrm{mem}}$. The same two-clock structure arises in both diffusion (DiT) and autoregressive (GPT) models, with architecture-dependent offsets. Dissecting the learned score of DiT models reveals a corresponding evolution of the optimization landscapes, where rule-valid samples' basins expand substantially around $τ_{\mathrm{rule}}$, while training samples' basins begin to dominate around $τ_{\mathrm{mem}}$. Together, these results yield a unified and predictive account of when and how generative models exhibit genuine innovation.

Authors: Amey Bhangale, Yezhou Zhang

Constraint satisfaction problems (CSPs) consist of a set of variables taking values from some finite domain and a set of local constraints on these variables. The objective is to find an assignment to the variables that maximizes the fraction of satisfied constraints. In this work, we study the CSP where the constraints are generalized linear equations over a finite group G. More specifically, for a given $S \subseteq G$, the constraints in this CSP are of the form addition of the values to the variables (similarly, product for non-abelian groups), belonging to the set $S$. We give an approximation algorithm for this problem on satisfiable instances and show that it is optimal for certain $S$ assuming $P\neq NP$. This natural predicate is one of the very few known predicates that are approximation resistant on almost satisfiable instances, assuming $P\neq NP$, but admits a non-trivial approximation algorithm on satisfiable instances.

Authors: Vahid R. Asadi, Atsuya Hasegawa, François Le Gall

It is known that there exist multi-prover interactive protocols ($\mathsf{MIP}$ protocols) for the complexity class $\mathsf{NEXP}$, succinct $\mathsf{MIP}$ protocols for $\mathsf{NP}$ and multi-prover interactive protocols with shared entanglement ($\mathsf{MIP}^\ast$ protocols) for $\mathsf{RE}$. This extraordinary power of multi-prover interactive proof systems comes from the assumption that provers do not communicate with each other during the protocols. If they are allowed to communicate freely, the setting is the same as in the single-prover case, and the computational power of the system becomes significantly weaker. In this paper, we investigate for the first time the setting where communication (i.e., leakage of information) between provers is allowed but bounded. We introduce two techniques to approach this question and show that multi-prover interactive proof systems are robust against some amount of leakage. Our first technique is based on parallel repetition theorems. We apply it to show that for any polynomial $p$, we can construct two-prover one-round $\mathsf{MIP}$ and $\mathsf{MIP}^\ast$ protocols for $\mathsf{NEXP}$ and $\mathsf{RE}$, respectively, that are robust against $p(n)$ bits of leakage. We further derive our second technique to convert any low-soundness PCP construction to a two-prover one-round $\mathsf{MIP}$ protocol for $\mathsf{NP}$ robust against leakage. We also discuss the relation between robustness against leakage in multi-prover interactive proof systems and the Sliding Scale Conjecture in the PCP literature.

Authors: Demian Banakh, Alexey Barsukov, Tamio-Vesa Nakajima

The logic MMSNP is a well-studied fragment of Existential Second-Order logic that, from a computational perspective, captures finite-domain Constraint Satisfaction Problems (CSPs) modulo polynomial-time reductions. At the same time, MMSNP contains many problems that are expressible as $ω$-categorical CSPs but not as finite-domain ones. We initiate the study of Promise MMSNP (PMMSNP), a promise analogue of MMSNP. We show that every PMMSNP problem is poly-time equivalent to a (finite-domain) Promise CSP (PCSP), thereby extending the classical MMSNP-CSP correspondence to the promise setting. We then investigate the complexity of PMMSNPs arising from forbidding monochromatic cliques, a class encompassing promise graph colouring problems. For this class, we obtain a full complexity classification conditional on the Rich 2-to-1 Conjecture, a recently proposed perfect-completeness surrogate of the Unique Games Conjecture. As a key intermediate step which may be of independent interest, we prove that it is NP-hard, under the Rich 2-to-1 Conjecture, to properly colour a uniform hypergraph even if it is promised to admit a colouring satisfying a certain technical condition called reconfigurability. This proof is an extension of the recent work of Braverman, Khot, Lifshitz and Minzer (Adv. Math. 2025). To illustrate the broad applicability of this theorem, we show that it implies most of the linearly-ordered colouring conjecture of Barto, Battistelli, and Berg (STACS 2021).

Authors: Michal Koucký, Ian Mertz, Sasha Sami

Catalytic computing concerns space bounded computation which starts with memory full of data that have to be restored by the end of the computation. Lossy catalytic computing, defined by Gupta et al. (2024) and fully characterized by Folkertsma et al. (ITCS 2025), is the study of allowing a small number of errors when resetting the catalytic tape at the end of a computation. Such a notion is useful when considering the robust use of catalytic techniques in the study of ordinary space-bounded algorithms. To that end however, defining and characterizing less strict notions of error was left open by Folkertsma et al. (ITCS 2025) and other works such as Mertz (B. EATCS, 2023). We expand the definition of possible resetting error in three natural ways: 1. randomized catalytic computation which can completely destroy the catalytic tape with some probability over the randomness 2. randomized catalytic computation which makes a bounded number of errors in expectation over the randomness 3. deterministic catalytic computation which makes a bounded number of errors in expectation over the initial catalytic tape itself We show a near complete characterization of the above models, both in the general case and in the logspace polynomial-time regime, by showing equivalences either between one another, to errorless catalytic space models, or to standard time or space complexity classes. Under a derandomization assumption, we show a near full collapse of all existing catalytic classes in the logspace regime.