Theory of Computing Report

Call for workshop proposals: FOCS 2026

Windows on Theory — 2026-06-17 02:31:07 UTC

[Guest post by Sam Hopkins]

We invite groups of interested researchers to submit workshop proposals for FOCS 2026 by July 31, 2026.

The FOCS 2026 workshops provide an informal forum for researchers to discuss important research questions, directions, and challenges in the field. Workshops often serve the vital purpose of introducing researchers to new areas and agendas. We also encourage workshops focusing on connections between theoretical computer science and other areas.

Format: Each workshop should be either 2.5 hours or 5 hours. All workshops are in person.

Proposal Submission

Workshop and tutorial proposals should fit within two pages. Please include a list of names and email addresses of the organizers, a brief description of the topic and the goals of the workshop, and the format (overview talks, invited talks, contributed talks, poster session, panel discussion, etc.) and proposed or tentatively confirmed speakers if known. The proposal should specify the desired duration (2.5 or 5 hours).

We encourage proposals to directly address how the topics will fit together and how the format will help researchers outside the particular subarea learn about it. If your proposal is accepted and you wish to solicit contributed talks, we can link to your call for contributions from the FOCS 2026 page. Feel free to contact the workshop co-chairs directly if you have any questions.

Co-chairs: Dakshita Khurana and Sam Hopkins

Submission Instructions

Proposals should be emailed to Dakshita Khurana (dakshita@illinois.edu) and Sam Hopkins (samhop@mit.edu) by July 31, 2026 anywhere on earth (AOE).

Proposers will be notified by August 14, 2026, whether their proposals have been accepted. Here is a sample template for workshop proposals; you may use a different format. The actual workshops will be held on November 8.

Call for workshops

By Boaz Barak

On the Complexity of the Circuit Width Problem

arXiv: Computational Complexity — 2026-06-17 00:00:00 UTC

Authors: Zhengfeng Ji, Yinchen Liu, Zhe'ou Zhou

Montanaro's polynomial representation expresses amplitudes of quantum circuits over the gates $H$, $Z$, $CZ$, and $CCZ$ as normalized gaps of degree-three polynomials over $\mathbb{F}_2$. The normalization is governed by the circuit width $w(f)$, the minimum number of qubits in any circuit realizing a polynomial $f$. Thus, efficient width minimization would give an approximate-counting route toward a combinatorial characterization of $BQP$. We study the computational complexity of this parameter. For degree-three polynomials with no constant term, deciding whether $w(f)\le k$ is $NP$-complete, resolving Montanaro's open question. We also prove $NP$-hardness of approximation within any factor $49/48-ε$, and show via a twin-copy construction that the exact and approximation hardness results also hold for degree-two polynomials. Under the Exponential Time Hypothesis, the exact problem admits no $2^{o(n)}$-time algorithm when $k=Θ(n)$. Complementing these hardness results, we give a nondeterministic polynomial-time search algorithm using $2\log_2\binom{n}{k}=O(k\log(en/k))$ witness bits, and a constructive fixed-parameter algorithm parameterized by $k$ with running time $k^{6k+o(k)}n+O(m)$.

Blended Chart Surfaces: A Seamless Explicit Representation for Smooth Surface Fitting

arXiv: Computational Geometry — 2026-06-17 00:00:00 UTC

Authors: Romy Williamson, Niloy Mitra

A surface representation suitable for geometry processing should be compact and explicit, provide global smoothness guarantees, support a wide range of surface topologies, and offer reliable access to differential quantities such as normals and surface energies, while remaining compatible with modern differentiable optimization. Existing neural representations typically sacrifice one or more of these properties: implicit fields typically require iso-surfacing for downstream use, while explicit neural maps are constrained by canonical-domain parametrizations or exhibit seam artifacts between local charts. We introduce Blended Chart Surfaces, a compact, network-free, explicit representation that is smooth by construction and anchored to user-provided topology. Given a coarse proxy mesh encoding the intended surface topology and approximate geometry, Blended Chart Surfaces jointly optimize for a polynomial map at each proxy vertex using an off-the-shelf optimizer to fit to an implicit target shape, avoiding the need for an input parametrization. Neighboring maps are fused using a smooth 'one-ring coordinate' blending scheme, decoupling topology and coarse geometry (carried by the proxy) from geometric details (carried by the local patches). The surface is globally smooth, fully differentiable, and enables stable evaluation of derivatives, making differential quantities and surface energies directly accessible. Additionally, our construction is equivariant to rigid motions and scaling of the proxy mesh. We evaluate Blended Chart Surfaces on various topologies and geometric complexity, and compare against explicit alternatives including interpolating-function baselines and mesh-displacement MLPs. Across these, Blended Chart Surfaces achieve a favorable trade-off among compactness, simplicity, access to differential quantities, and expressivity while remaining smooth across patch boundaries.

Greedy Vector Balancing

arXiv: Computational Geometry — 2026-06-17 00:00:00 UTC

Authors: Wojciech Czerwiński, Daniel Dadush, Ekin Ergen, Arka Ghosh, Sławomir Lasota, Łukasz Orlikowski

In online vector balancing, vectors $t_1,\dots,t_n$ arrive one by one from a given set $T$ and the goal is to assign signs $s_1,\dots,s_n\in\{\pm1\}$ in an online manner so as to minimize the largest norm of any signed prefix sum $\sum_{i=1}^ks_i t_i$, $k \in [n]$. In this paper, we analyze the natural Euclidean greedy vector balancing algorithm for this problem: at each step $k$, the sign $s_k\in\{\pm1\}$ is chosen so that $s_k t_k$ has non-positive inner product with $\sum_{i=1}^{k-1} s_i\cdot t_i$. Our main result is the first finite bound, independent of the sequence length $n$, on the performance of greedy whenever $T$ is finite. When $T \subset \mathbb{R}^d$ consists of unit vectors, we prove that the signed sums produced by greedy have Euclidean norm at most $(2/δ_T)^{d-1}$, where $δ_T$ is the minimum non-zero distance between vectors in $T$ and subspaces spanned by vectors in $T$. The same upper bound holds when the sequences are composed of scaled down vectors in $T$. We also provide a simple set $T$ for which $Ω(\sqrt{d}/δ_T)$ is a lower bound. We analyze the greedy algorithm by proving the existence of a bounded convex $K_T$ that is $T$-absorbing: $\forall x\in K_T$ and $t \in\pm T$, $\langle x,t\rangle\leq0\Rightarrow x+t\in K_T$. We give an explicit construction of a set $K_T$ contained in a ball of radius $(2/δ_T)^{d-1}$, based on chains of subspaces spanned by vectors in $T$, which may be of independent interest. We generalize our greedy vector balancing bound to online vector partitioning, where the sequence $t_1,\dots,t_n$ must be partitioned in an online manner into $p$ subsequences. As an application, we prove a special case of a conjecture of Bosman et al. (arxiv:2402.19259), showing that a lexicographic version of total completion time scheduling under scenarios is polynomial time solvable when the number of scenarios is fixed.

High-Fidelity 3D Geometric Reconstruction of Pelvic Organs from MRI: A Hybrid Deep Learning and Iterative Optimization Approach

arXiv: Computational Geometry — 2026-06-17 00:00:00 UTC

Authors: Hui Wang, Xiaowei Li, Chenxin Zhang, Yifan Feng, Jianwei Zuo, Yumeng Tang, Xiuli Sun, Jianliu Wang, Bing Xie, Jiajia Luo

Patient-specific 3D reconstruction of pelvic organ geometry from MRI is important for pelvic floor modeling and downstream patient-specific analysis. However, while previous studies have focused primarily on either image segmentation or downstream use of 3D models, the reconstruction of high-fidelity, high-quality geometries remains labor-intensive and poorly standardized. The study introduced a hybrid deformable shape modeling framework that integrates deep learning prediction with iterative optimization for the reconstruction of the bladder, uterus, and rectum. The framework consists of three core components: a geometry-aware multi-level deep learning architecture that preserves topological consistency of pelvic organs; a two-stage amortized optimization training strategy that balances global shape capture and local surface refinement; and a holistic synergy mechanism--where iterative optimization provides supervision for deep learning during the training phase, and during inference, deep learning rapidly predicts the global organ morphology, followed by iterative optimization to refine local surfaces and mesh quality. This framework demonstrated marked superiority in geometric fidelity than current mainstream deep learning-based organ reconstruction models. For individual anatomical structures, the reconstructed 3D geometries for the bladder, rectum, and uterus achieved significantly lower Chamfer Distance values and higher Dice Similarity Coefficient scores. In addition, while maintaining high computational efficiency, the proposed architecture yielded superior overall volumetric mesh quality. At the patient level, the framework achieved higher mean values for the 10 worst elements for both minSICN and minSIGE compared to traditional geometric post-processing algorithms.

Non-negative Matrix Factorisation with Topological Regularisation

arXiv: Computational Geometry — 2026-06-17 00:00:00 UTC

Authors: Matias de Jong van Lier, Shizuo Kaji, Keunsu Kim

We investigate the learning of interpretable bases in non-negative matrix factorisation (NMF) by regularising the topology of the learned basis functions. Our approach is motivated by the observation that many data modalities can be viewed as non-negative functions on a structured domain, where the quality of a basis is intrinsically linked to its topology. However, naive methods for incorporating the topology of the support are often hindered by discreteness and threshold dependence, rendering them unsuitable for continuous optimisation. We address these challenges by employing persistent homology as a stable, threshold-free topological quantifier and by designing topological scores that integrate into the NMF objective as regularisers. The resulting framework encompasses spatially coherent image components, periodic time-series structures, and clique-like graph signals within a unified modelling language.

Agent Utilities over Generalized Voronoi Regions and their Gradients

arXiv: Computational Geometry — 2026-06-17 00:00:00 UTC

Authors: Andre N. Costa, Petter Ögren, Carlos H. C. Ribeiro

In this paper, we generalize the concept of Voronoi regions, define agent utility as the integral of a utility density over the corresponding Voronoi region, derive gradients of the utility, and illustrate the approach in a two-team example from soccer. The generalization of Voronoi regions is in the form of so-called Cost-Induced Voronoi (CIV) regions, where the agent state space may differ from the space being partitioned. One example of such regions is when the cost is given by the optimal solution of an LQR control problem. Then the agent states include position as well as velocity, while the partitioned space only includes positions. The agent utility is defined by integrating some utility density over the CIV region of the agent. This utility density might be the probability density of some beneficial event, such as receiving a pass in soccer. The utility is then the overall probability of receiving a pass and the gradient represents a way to improve that probability. We show how this utility gradient can be computed using the Reynolds Transport Theorem from fluid mechanics, and that this approach achieves similar accuracy while reducing computation time by about an order of magnitude compared to a baseline finite-difference approximation.

Directed Reachability-Preserving Minimum Edge Cut: Approximation and Planar Hardness

arXiv: Data Structures and Algorithms — 2026-06-17 00:00:00 UTC

Authors: Qi Duan

We study a directed version of the three-terminal reachability-preserving minimum edge cut problem. Given a directed graph $G=(V,A)$ with arc costs and terminals $s_1,s_2,t$, the one-way directed RPMEC problem asks for a minimum-cost set of arcs whose deletion preserves the reachability $s_1\leadsto s_2$ while destroying the reachability $s_1\leadsto t$. We first give a path--cut formulation in terms of a rooted directed cut function. Using a root-linear approximation for the associated polymatroid, we obtain an $O(\sqrt r)$-approximation, where $r$ is the number of relevant vertices with positive singleton cut value. In particular this gives an $O(\sqrt n)$-approximation in general directed graphs. For acyclic directed graphs, we give an additional singleton-length algorithm and obtain an $O(\min\{\sqrt r,h\})$ guarantee, where $h$ is the maximum number of relevant vertices on an $s_1$-$s_2$ path. Finally, we prove that directed planar RPMEC is NP-hard, even on acyclic planar digraphs with nonnegative costs, by reducing from independent set on cubic planar graphs through a finite-bimodal directed node-cut construction and a planar node-to-edge split.

A Counterexample to Wegner's Conjecture for Axis-Parallel Rectangles

arXiv: Data Structures and Algorithms — 2026-06-17 00:00:00 UTC

Authors: Deepak Ajwani, Rishikesh Gajjala, Rajiv Raman, Saurabh Ray

In 1965, Wegner conjectured that every finite family $\mathcal R$ of axis-parallel rectangles in the plane satisfies $τ(\mathcal R) \le 2ν(\mathcal R)-1$, where $τ(\mathcal R)$ denotes the minimum number of points needed to pierce all rectangles in $\mathcal R$, and $ν(\mathcal R)$ denotes the maximum size of a pairwise disjoint subfamily. Over the last six decades, the conjecture has motivated a long line of work: it has been verified for several special classes of rectangle families, and the best known general upper bounds have been progressively improved, but the conjecture itself had remained open. We give an explicit counterexample. More precisely, we construct a triangle-free rectangle-intersection graph on $n$ vertices whose independence number is at most $n/4$. Since the graph is triangle-free, no point of the plane can lie in three rectangles; hence every piercing point hits at most two rectangles. Consequently, $τ(\mathcal R) \ge n/2 \ge 2ν(\mathcal R)$, contradicting Wegner's conjectured bound. We also give a slightly more general construction for which $τ(\mathcal R) \ge 2.21ν(\mathcal R)$. This shows that the standard point relaxation, equivalently the clique relaxation, for the Maximum Independent Set of Rectangles problem has integrality gap at least $2.21$.

Online Connectivity Augmentation

arXiv: Data Structures and Algorithms — 2026-06-17 00:00:00 UTC

Authors: Mohit Garg, Aditya Subramanian

The Connectivity Augmentation Problem (CAP) is a fundamental problem in fault-tolerant network design and has been extensively studied in the context of approximation algorithms. In this work, we consider CAP in the online setting: given a $k$-edge-connected graph $G$ and a set $L$ of additional edges over the vertices of $G$, called links, online requests arrive one by one, each specifying two vertices that need to be $(k+1)$-edge-connected. We start with the graph $G$ and progressively add links to serve these requests. More specifically, upon the arrival of a request $\{u,v\}$, we must immediately and irrevocably add zero or more links from $L$ to the graph so that $u$ and $v$ are $(k+1)$-edge-connected in the resulting augmented graph. The goal is to minimize the total number of links added, and we evaluate an algorithm's performance by its competitive ratio relative to an optimal offline solution. In this work, improving upon previous bounds, we obtain a tight competitive ratio for online CAP, along with other related results.

Reducing Prize-Collecting Stroll and Related Routing Problems to Prize-Collecting TSP

arXiv: Data Structures and Algorithms — 2026-06-17 00:00:00 UTC

Authors: Hong Li

The prize-collecting stroll is the path version of the prize-collecting TSP. Given a complete metric graph, two prescribed terminal vertices $s,t$, and nonnegative penalties on vertices, the prize-collecting stroll asks for an $s$-$t$ tour minimizing the length of the tour plus the total penalty of vertices that are not visited by the $s,t$ tour. We study a common generalization of the prize-collecting stroll and several related prize-collecting routing problems, which we call the prize-collecting-$Φ$-TSP. In this model, $Φ$ specifies a set of prescribed vertices alongside their parity and connectivity requirements. We show that, if a $ρ$-approximation algorithm for the prize-collecting TSP is available, then, for any $\varepsilon>0$, there is a polynomial time $(ρ+\varepsilon)$-approximation algorithm for the prize-collecting-$Φ$-TSP when the number of prescribed vertices is bounded by a fixed constant. This yields a better-than-$1.6$-approximation algorithm for the prize-collecting stroll, improving the previous best-known approximation guarantee of $1.6662$.

A polynomial-time approximation scheme for minimum-weight decoding of topological codes

arXiv: Data Structures and Algorithms — 2026-06-17 00:00:00 UTC

Authors: Shouzhen Gu, Lily Wang, Aleksander Kubica

Two-dimensional topological translationally invariant (2D TTI) stabilizer codes lie at the heart of fault-tolerant quantum computation, but using them requires solving the decoding problem. Minimum-weight decoding of these codes was recently shown to be NP-hard, even in basic settings, such as the color code with Pauli $Z$ errors and the toric code with Pauli $X$, $Y$ and $Z$ errors. Here, we prove that minimum-weight decoding of 2D TTI codes nonetheless admits a polynomial-time approximation scheme (PTAS), i.e., for any constant $\varepsilon>0$, a recovery operator of weight within a multiplicative factor of $1+\varepsilon$ of the minimum can be found in polynomial time. Our approach builds on Arora's PTAS for Euclidean problems, such as the traveling salesman problem, and applies when decoding can be cast in terms of point-like excitations connected by string-like errors. It therefore extends beyond two dimensions, covering certain higher-dimensional topological codes and quantum memories, including the toric code with phenomenological or circuit-level noise.

The Value of Adaptivity in LSM Bloom-Filter Tuning: A Log-Law and a Two-Clock Frontier

arXiv: Data Structures and Algorithms — 2026-06-17 00:00:00 UTC

Authors: Madhulatha Mandarapu, Sandeep Kunkunuru

Log-structured merge (LSM) trees attach an approximate-membership filter to every run and must split a fixed memory budget across them. The static optimum is known (Monkey); a large systems literature then makes the allocation adaptive, tracking shifting hotness online. We ask a prior question: when is that adaptivity worth its machinery? We give three analytical answers and validate them on synthetic sweeps, real Twitter production cache traces, and a real RocksDB engine. First, a log-law: optimal bits-per-key is affine in the logarithm of access frequency, at a fixed slope. Second, a robustness law: because the workload enters only logarithmically, the excess read cost from a hotness misestimate is half the size-weighted variance of the log error, and a common-factor misestimate is absorbed by the budget multiplier, so coarse estimates lose little. Third, an adaptivity-value frontier: since compaction rebuilds filters for free on its own clock, the value of continuous tracking over an allocation recomputed only at compaction grows quadratically in the within-epoch drift, with a closed-form scale. This yields a three-regime policy (coarse-at-compaction suffices, then track, then at extreme drift fall back to uniform) and predicts that more skew makes fine tracking matter less. On a real cluster, reallocating only at compaction captures 96-99% of tracking's benefit; on RocksDB the false-positive primitive holds within four percent to eight bits per key. The contribution is a characterization of when adaptive tuning pays; we add no new filter and no engine fork. Code and pre-registration are public.

The independence number of uncrowded hypergraphs: bounds matching the shattering threshold

arXiv: Data Structures and Algorithms — 2026-06-17 00:00:00 UTC

Authors: Abhishek Dhawan, Abhishek Methuku, Minh-Quan Vo

A foundational theorem of Ajtai, Komlós, Pintz, Spencer, and Szemerédi asserts that every $n$-vertex $k$-uniform uncrowded hypergraph with maximum degree $Δ$ contains an independent set of size $c_k n{\left(\frac{\log Δ}Δ\right)^{\frac{1}{k-1}}}$, for some constant $c_k>0$. Determining the optimal leading constant $c_k$ in this bound is a major open problem. A natural target is the so-called shattering-threshold constant $\left(\frac{1}{k-1}\right)^{\frac{1}{k-1}}$, which appears in the solution-space geometry of random constraint satisfaction problems, in average-case complexity theory, and in statistical physics, among other areas. We prove that uncrowded hypergraphs attain this threshold. More precisely, for every $ε>0$ and $k\geq 2$, every $n$-vertex $k$-uniform uncrowded hypergraph of sufficiently large maximum degree $Δ$ contains an independent set of size at least $(1-ε) n {\left(\frac{1}{k-1}\frac{\log Δ}Δ\right)^{\frac{1}{k-1}}}$. Consequently, we obtain the first pseudorandom class of hypergraphs whose guaranteed independence number matches the shattering threshold, resolving a folklore conjecture. Moreover, as another direct consequence, we resolve a conjecture of Verstraëte and Wilson by proving that there exists a constant $c_k=1-o_k(1)$ such that every $n$-vertex $k$-uniform linear hypergraph of maximum degree $Δ$ has independence number at least $c_k n\left(\frac{\log Δ}Δ\right)^{\frac{1}{k-1}}$. Our techniques are constructive yielding efficient algorithms for both static and distributed settings. Specifically, we provide an $\tilde O(nΔ)$-time randomized static algorithm and an $\tilde O(1)$-round randomized $\textsf{LOCAL}$ algorithm to find an independent set in uncrowded hypergraphs at the shattering threshold. These results extend seamlessly to the the setting of linear hypergraphs.

Four-Cycle Counting in Low-Degeneracy Graph Streams

arXiv: Data Structures and Algorithms — 2026-06-17 00:00:00 UTC

Authors: Sebastian Lüderssen, Stefan Neumann, Pan Peng

We study the problem of $(1+\varepsilon)$-approximating the number of four-cycles in graphs given as arbitrary order edge streams. We propose two new algorithms based on sampling induced subgraphs. Our first contribution is a two-pass algorithm that uses $\widetilde{O}(κm / \sqrt{T})$ space, where $m$ is the number of edges, $T$ is the number of four-cycles, and $κ$ is the graph's degeneracy. This algorithm improves upon existing theoretical bounds and is provably optimal for constant-degeneracy graphs, matching the known $Ω(m/\sqrt{T})$ lower bound up to lower-order factors. Our second contribution is a one-pass algorithm that remains accurate when four-cycles are not highly concentrated around individual nodes, edges, or wedges; this structural property is common in sparse social and collaboration networks. We evaluate both algorithms on a variety of real-world graph streams. The two-pass algorithm consistently outperforms state-of-the-art methods, using substantially less space to achieve a desired accuracy. The one-pass algorithm is competitive when four-cycles are evenly distributed, matching our theoretical analysis. Unlike several recent works, our algorithms perform well even on non-bipartite graphs such as social networks.

The 2026 Algorithmic Information Theory Data Compression Challenge

arXiv: Data Structures and Algorithms — 2026-06-17 00:00:00 UTC

Authors: André Ribeiro, Rúben Garrido, Violeta Ramos, António Alberto, Diogo Fernandes, João Varela, Eduardo Lopes, Rodrigo Abreu, Hugo Ribeiro, Tomás Brás, David Pelicano, Afonso Ferreira, Sebastião Teixeira, Maria Linhares, Martim Santos, Rui Machado, Duarte Santos, Gabriel Silva, Guilherme Rosa, João Roldão, Henrique Teixeira, Cláudia Seabra, Ricardo Fonseca, Richard Miranda, Hugo Castro, Ângela Ribeiro, Fouad Bellili, Luís Diogo, André Cardoso, Armando J. Pinho, Diogo Pratas

Lossless data compression remains central to computer science, with direct impact on storage, communication bandwidth, computational cost, and energy consumption. It is also closely related to Algorithmic Information Theory, where compressibility provides an operational measure of structure and non-randomness. This paper presents the 2026 Algorithmic Information Theory Data Compression Challenge, a benchmark for evaluating general-purpose lossless compressors under realistic constraints. Submissions were encouraged to use arithmetic or range coding, limited to at most 8 GB of memory, and required to include a decompressor no larger than 1 MB. The benchmark comprised sixteen heterogeneous files, split into public training and hidden testing datasets. In total, 117 valid submitted compressors were evaluated alongside established reference compressors using compression ratio, compression and decompression time, Weissman score, and Pareto-frontier analysis. The results show that performance depends strongly on the optimization criterion: fast compressors achieved the best speed-oriented scores, whereas modelling-intensive compressors produced smaller outputs at higher computational cost. A Normalized Compression Distance analysis further revealed clusters of related submissions and distinguished incremental variants from more independent implementations. Selected submissions were described for their methodological novelty or competitive performance and further tested on four large external datasets, where several achieved competitive or superior results relative to established compressors. Overall, the challenge confirms the importance of probabilistic modelling, hidden testing, and external datasets for assessing compression performance and generalization. Benchmark resources, leaderboard data, binaries, and selected source code are publicly available at https://aitdcc.github.io.

Spectral recovery of a planted triangle-dense subgraph

arXiv: Data Structures and Algorithms — 2026-06-17 00:00:00 UTC

Authors: Sam van der Poel, Cheng Mao, Benjamin McKenna

Given a simple graph on $n$ vertices and a parameter $k$, the triangle-densest-$k$-subgraph problem is known to be computationally hard in the worst case. To circumvent the computational hardness, we study an average-case model where a triangle-dense subgraph on $k$ vertices is planted in an Erdős-Rényi random graph on $n$ vertices. For the recovery of the planted subgraph, we propose a simple spectral algorithm and a semidefinite program, both of which use a graph matrix whose entries are local signed triangle counts. Theoretical guarantees for these algorithms are established through spectral analysis of the graph matrix. Finally, we provide evidence showing a statistical-to-computational gap analogous to that for the planted clique problem. The computational threshold in terms of the subgraph size $k$ is at least $\sqrt{n}$ in the framework of low-degree polynomial algorithms, while the information-theoretic threshold is at most logarithmic in $n$.

Exact Algorithms for Edge Deletion to Cactus Graphs and Weighted Variants

arXiv: Data Structures and Algorithms — 2026-06-17 00:00:00 UTC

Authors: Wenhao Song

We study exact exponential-time algorithms for Edge Deletion to Cactus. Given a connected graph $G$, the task is to delete a minimum number of edges so that the remaining spanning graph is a connected cactus. Akhtar and Philip (IWOCA 2026) gave an $O^*(3^n)$-time algorithm for the unweighted problem, where $n$ is the number of vertices in the input graph and the $O^*(\cdot)$ notation hides polynomial factors. We improve this bound to $O^*(2^n)$ time and space. More generally, if the deletion costs take at most $q$ distinct nonnegative real values, then the weighted problem can be solved in $O^*(2^n n^{O(q)})$ time and space. Thus every fixed number of distinct costs, and in particular the unweighted case, admits a faster exact algorithm. For nonnegative integer costs of total weight $W$, we obtain an $O^*(2^n(W+1))$ pseudo-polynomial algorithm, while arbitrary nonnegative real costs admit an $O^*(3^n)$ exact algorithm.

Approximation Preserving Coresets

arXiv: Data Structures and Algorithms — 2026-06-17 00:00:00 UTC

Authors: Milind Prabhu, Chris Schwiegelshohn, Sudarshan Shyam

Clustering in a big data setting is an intensively studied problem, with coresets emerging as one of the important paradigms in this line of work. Given a cost function $\text{cost}(P,S)$ mapping input points $P$ and a solution $S$ to an objective value, a coreset is a typically weighted sketch $Ω\subseteq P$ such that $\text{cost}(Ω,S)\approx \text{cost}(P,S)$. In practice, coreset sizes much smaller than those suggested by theoretical guarantees are often found to be sufficient. In this paper, we offer an explanation for this phenomenon. Smaller coreset sizes suffice if we only wish to preserve the costs of \emph{good} solutions, i.e., solutions with low cost. We define and devise \emph{approximation-preserving coresets}, which provide a weaker guarantee than strong coresets, which apply to all solutions, while providing stronger guarantees than weak coresets, which apply only to the optimum solution. We complement this result by showing that even a very small distortion in the approximation factor cannot admit coresets of this size.

Space-Efficient Lock-Free Linear-Probing Hash Table

arXiv: Data Structures and Algorithms — 2026-06-17 00:00:00 UTC

Authors: Hagit Attiya, Rotem Oshman, Noa Schiller

Linear probing is one of the simplest and most space-efficient approaches to hash table design, and is widely used in sequential settings due to its compact memory layout. However, designing a concurrent linear-probing hash table with strong liveness guarantees has proved difficult, and only a handful of such algorithms have been proposed, all of which either restrict concurrency or rely on large per-entry metadata, thereby compromising space efficiency. We present a lock-free linear-probing hash table with wait-free lookups that retains the core advantages of sequential linear probing while handling contention gracefully. Our design uses only a small amount of metadata per table entry: a constant number of additional bits when using LL/SC, or a logarithmic number of bits when using CAS. The algorithm is linearizable and lock-free, supports insert, delete, and wait-free lookup operations, and is able to safely reclaim space used by deleted elements without rebuilding the table. We analyze the amortized step complexity of our hash table assuming no concurrent insertions of the same key, and show that each operation has expected amortized step complexity matching that of sequential linear probing, up to the point contention per key.

Scalable K-clique Estimation with Differential Privacy

arXiv: Data Structures and Algorithms — 2026-06-17 00:00:00 UTC

Authors: Dung Nguyen, Ritwick Mishra, Anil Vullikanti

Counts of $k$-cliques are commonly used metrics in subgraph mining. Since graphs often have sensitive data, there also has been a lot of work on $k$-clique counts with differential privacy. However, these metrics have very high global sensitivity, and so more sophisticated techniques have been developed for counting $k$-cliques with privacy. Smooth sensitivity and ladder functions were developed for reducing the noise magnitude for private estimates of these metrics. However, these are computationally very inefficient to estimate. No polynomial time algorithms are known for smooth sensitivity of $k$-cliques for $k>3$, while the time complexity of ladder functions is lower bounded by the time for exact counts, which does not scale very well. In this paper, we develop a new highly scalable algorithm for estimating $k$-clique counts with differential privacy. Our algorithm adapts the ladder function to serve as a smooth upper bound on its local sensitivity, and utilizes the approximation sensitivity framework to calibrate noise with magnitude proportional to an approximation of the bound. This gives us a significant improvement in the running time. Experiments show that our method is several orders of magnitude faster than the ladder function based estimates of $k$-clique counts, while the accuracy is similar. Our algorithm is the first to scale to graphs with millions of edges, and for larger $k$, for which the ladder function algorithm doesn't complete.

Adaptive Proximal Methods for Weakly Convex Optimization with Unknown Parameter: Deterministic and Stochastic Guarantees

arXiv: Data Structures and Algorithms — 2026-06-17 00:00:00 UTC

Authors: Miaolan Xie

Many nonsmooth, nonconvex objectives in learning and signal recovery are $ρ$-weakly convex. We minimize such a function in deterministic and stochastic settings when the weak-convexity parameter $ρ$ is unknown. The objective is not required to be globally Lipschitz continuous or smooth. We propose the Adaptive Prox-Guided Scheme (APS), a one-trial proximal algorithm that adapts the proximal parameter online and bidirectionally through a descent test, allowing it to exploit favorable local structure. In the deterministic setting, APS obtains an $O(\varepsilon^{-2})$ iteration complexity for producing an $\varepsilon$-subgradient stationary point. In the stochastic setting, APS achieves a high-probability $O(\varepsilon^{-2})$ iteration bound for driving the Moreau-envelope gradient below $\varepsilon$. This result holds under deliberately weak oracle assumptions: the function-difference estimates may be biased and heavy-tailed, and the stochastic proximal oracle need only be sufficiently accurate with constant probability when the proximal parameter lies below $1/(2ρ)$ (unknown to the algorithm), and can be arbitrary otherwise.

Sum-of-Squares Degree Barriers for the Reweighted-Hinge Method in Robust Halfspace Learning: A Christoffel-Function Characterization

arXiv: Data Structures and Algorithms — 2026-06-17 00:00:00 UTC

Authors: Xiaoyu Li

A certificate that removes outliers sees the data only through its low-degree moments, and an adversary exploits exactly this, hiding corruption where the clean data already looks typical, in the blind spot no bounded-degree test resolves. That blind spot turns out to have an exact size: the Christoffel function of the clean marginal, the very quantity modern data analysis thresholds to detect outliers, here read from the adversary's side as the corruption a bounded-degree certificate cannot remove. We turn this inversion into the organizing principle of the reweighted-hinge approach to robustly learning $γ$-margin halfspaces under malicious noise (Shen, 2025; Zeng and Shen, 2025): the governing resource is the Sum-of-Squares degree of the outlier-removal certificate, and the resolution principle states that the maximal corruption mass which can hide at a center $c$ from a degree-$2t$ certificate is exactly the Christoffel function $λ_{t+1}(c)$ of the clean marginal. Three consequences follow, all against the certificate method (not information-theoretic). A margin-degree tradeoff: certifying the dense pancake to error $ε$ costs SoS degree $Ω(\log(1/ε))$ or margin $Ω(\sqrt{\log(1/ε)}/\sqrt{d})$, explaining why the $\log(1/ε)$ margin Shen (2025) records is forced, with a weighted-Chebyshev reduction making the threshold $2t=Θ((|c|/s)^2)$ tight modulo one classical weighted-extremal estimate. A degree-$2$ outlier barrier: the resolution principle realized as an explicit instance on which degree $2$ is stuck at $η^{1/2}$ while degree $4$ escapes, locating the method's small breakdown rate in the degree, not the analysis. And a degree-$2t$ algorithm tracing the frontier $η^{1-1/2t}$ (recovering Shen (2025) at $t=1$), whose gain is an explicit constant, capped by the pancake density and shown unimprovable by the degree-$2$ barrier.

The Complexity of Min-Max Optimization for Quadratic Polynomials

arXiv: Computational Complexity — 2026-06-16 00:00:00 UTC

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

We prove that computing approximate stationary points of min-max optimization over the hypercube is PPAD-hard for quadratic polynomials. This holds even when the polynomials are multilinear, each variable appears in at most three monomials, and the approximation factor is inverse polynomial. As a direct consequence, we obtain the first PPAD-hardness results for two-team zero-sum polymatrix games.

Worst-case depth hierarchy for shallow quantum circuits

arXiv: Computational Complexity — 2026-06-16 00:00:00 UTC

Authors: Min-Hsiu Hsieh, Michael de Oliveira, Sathyawageeswar Subramanian, Xingjian Zhang

Circuit depth is a central resource in complexity theory. While bounded-depth classical circuits admit well-understood hierarchy theorems, the internal structure of constant-depth quantum computation remains comparatively unexplored. We prove an explicit depth hierarchy theorem for $\mathsf{QNC}^0$. For each $d\ge 12$, we construct a family of two-round interactive problems on which no depth-$(d-1)$ quantum circuit can achieve near-perfect success, regardless of gate set, circuit size, or ancillary qubits. In contrast, we prove that our construction admits realizations by simple bounded fan-in quantum circuits of depth larger than $d$ by a small constant factor. Moreover, all bounded fan-in classical circuits of sublogarithmic depth (in the input size) fail to achieve perfect success on these tasks for every $d$, yielding a hierarchy of problems that show unconditional quantum advantage of $\mathsf{QNC}^0$ over $\mathsf{NC}^0$. A key obstacle is the scarcity of lower bound techniques for quantum circuits. To address this, we develop methods to analyze how depth affects a circuit's ability to realize nonlocal correlations amongst its output qubits in a fine-grained manner. Our approach exploits the correspondence between constraint systems and nonlocal games, translating group-theoretic constructions into rigid operator-valued constraint systems and then into non-local games. In particular, we construct constraint systems whose unique faithful operator-valued solutions require every perfect strategy, and every near-perfect strategy to a fixed precision, to implement multi-controlled phase operations. This reduces to a nonlocal unitary-synthesis problem, yielding depth lower bounds for both shallow quantum and classical circuits. These results show that increasing depth strictly increases computational power within $\mathsf{QNC}^0$, establishing a genuinely quantum hierarchy.

The Computational Complexity of Team Zero-Sum Games

arXiv: Computational Complexity — 2026-06-16 00:00:00 UTC

Authors: Ioannis Anagnostides, Ioannis Panageas, Tuomas Sandholm, Jingming Yan

A celebrated consequence of the minimax theorem is that two-player zero-sum games admit a tractable equilibrium characterization. In many central applications, however, each side comprises multiple independent agents who share a common objective but cannot perfectly coordinate their actions. Such settings can be modeled as \emph{team zero-sum games}, a natural generalization of both two-player zero-sum games and potential games -- the two most well-studied classes of games in algorithmic game theory. In this paper, we settle the complexity of team zero-sum games by establishing that computing Nash equilibria is \PPAD-complete. As a result, despite the global adversarial structure, team zero-sum games are as hard as general-sum games. Our hardness result holds even when i) the precision is inverse polynomial, thereby ruling out a fully polynomial-time approximation scheme (unless $¶= \PPAD$); ii) each team consists of only two players; and iii) the underlying class of games is polymatrix. As a byproduct, we resolve the complexity of group-wise zero-sum polymatrix games, a class introduced and examined in the seminal work of Cai and Daskalakis (SODA '11), and more recently highlighted by Hollender, Maystre, and Nagarajan (ICLR '25). Moreover, we show that computing a first-order stationary point in min-max optimization is \PPAD-complete even for quadratic (multilinear) objectives. From a technical standpoint, we develop a series of team zero-sum game gadgets that allow us to simulate the breakthrough reduction of Bernasconi and Castiglioni (STOC '26). Moreover, to obtain hardness results for quadratic objectives, we make use of a general technique based on linear local approximation, which is of independent interest.

Revisiting average case complexity of multilevel syllogistic: From the 1995 Courant Technical Report to Lean 4 Formalization

arXiv: Computational Complexity — 2026-06-16 00:00:00 UTC

Authors: Lars Warren Ericson

We describe a Lean~4 formalization revisiting NYU Courant Technical Report TR1995-711 on the average-case complexity of Multilevel Syllogistic (MLS). The development encodes Reischuk--Schindelhauer average-case classes, an axiomatic MLS/EMLS semantics layer, a partial Ferro--Omodeo--Schwartz decision procedure with proved soundness and partial completeness on a membership-free fragment, serialization and step budgets, and conditional NP-average completeness and non-AvP hardness corollaries modulo explicitly documented structural axioms. Full Lean sources are inlined in the appendix modules.

Polynomial-Time Mistake-Bounded Language Generation

arXiv: Computational Complexity — 2026-06-16 00:00:00 UTC

Authors: Héctor Jimenez, Alexander Kozachinskiy, Vicente Opazo

In this note, we introduce a polynomial-time version of the mistake-bounded language generation (MBLG) framework due to Kleinberg, Peale, and Reingold (2026). We observe that the family of parities of variables, and the family of conjunctions of literals, are polynomial-time MBLG. Our main result states that the family of monotone Boolean functions with polynomially-many maxterms is polynomial-time MBLG. This family includes all monotone Boolean functions, computable by polynomial-size decision trees. Our technique can be presented as a new combinatorial game about writing numbers on a board.

Quiet Planting for $k$-SAT, Multiple Solutions of Arbitrary Geometry

arXiv: Computational Complexity — 2026-06-16 00:00:00 UTC

Authors: Ali Ahmadi, Kiarash Banihashem, Iman Gholami, Mohammad Taghi Hajiaghayi, Jan Olkowski

Recent work on "quiet planting" in combinatorial optimization aims to generate instances with a hidden solution that is hard to recover, typically by making the planted distribution statistically indistinguishable from uniform for specific algorithms, such as statistical queries. A prominent example is planted $k$-SAT, where $O(n^{k/2})$ clauses can be planted while maintaining indistinguishability from uniform instances, evidenced by prior hardness results which also align with findings in SAT refutation. Despite extensive research and practical use in benchmarking SAT solvers, the challenge of quietly planting multiple solutions while preserving hardness has remained an open problem. This work initiates the study of quiet planting with an arbitrary number of solutions, proposing the first method to construct quiet planting distributions for $k$-SAT formulas that accommodate more than one solution. We provide statistical query lower bounds for distinguishing these planted instances from uniform ones, and our method allows for planting solutions with arbitrary geometric relationships, including varying Hamming distances. A key innovation facilitating multiple solutions is the ability to incorporate arbitrary correlations between variable selection in clauses and their negation patterns, departing from prior approaches. We also investigate the worst-case complexity of SAT by showing the difficulty in distinguishing satisfiable instances with numerous solutions from unsatisfiable ones, addressing an open problem of Hsieh, Mohanty, and Xu (CCC'22). Technically, we generalize $(r-1)$-wise uniformness in clause distributions, proving hardness if marginal negation distributions are $(r-1)$-wise uniform. We also reveal a connection to binary linear codes, showing a $[k, t, r]$ code can guide planting up to $2^t - 1$ solutions on $k$ variables.

The Exact Reach of Conormal Invariants in Determinantal Complexity: a Quadratic No-Go Theorem

arXiv: Computational Complexity — 2026-06-16 00:00:00 UTC

Authors: Karthik Sheshadri

We study the polar (conormal) method for determinantal-complexity lower bounds, including the framework used in the companion bound dc(sum_i x_i^N) >= (1/(4e)-o(1))N^2. We obtain quantitative results on both sides of the method: the intersection-theoretic complexity of kernel-incidence constructions and the size of the characteristic-cycle invariants they can detect. For a size-m determinantal representation in N variables, we identify the corank-one kernel incidence with the conormal variety of the generic determinant. An excess-one degeneracy-locus computation yields a closed formula for the associated polar intersection number T(N,m), together with rational generating functions and explicit evaluations including T(3,m)=m(m-1), T(4,m)=m(m-1)^2, and T(5,5)=220. We also compare these counts with the multihomogeneous Bezout estimates used in the companion work and establish asymptotic sharpness at the per-root scale. For an arbitrary degree-d hypersurface X in P^(N-1), possibly singular and reducible, we prove a uniform bound on the conormal multidegrees appearing in its characteristic cycle: m_S delta_i(Con(S-bar)) <= 8(d-1)^(N-1)+O(N). The proof combines bounds on generic-slice Euler characteristics, an explicit analysis of the transform from Euler-characteristic data to conormal multidegrees, and Kashiwara positivity for characteristic cycles. Similar bounds are obtained for vanishing-cycle and Milnor-class variants. Combining these results yields general upper bounds on determinantal-complexity lower bounds obtainable from characteristic-cycle invariants via kernel-corank incidence constructions. In particular, along the diagonal d=N, the resulting lower bounds are at most quadratic in N. We conclude by discussing possible extensions involving scheme-theoretic conormal information and other geometric invariants.

Three-Terminal Reachability-Preserving Minimum Node Cut: Planar Hardness and a General-Graph $O(\sqrt n)$-Approximation

arXiv: Computational Complexity — 2026-06-16 00:00:00 UTC

Authors: Qi Duan

We study the three-terminal reachability-preserving minimum node cut problem (\RPMNC). The input is an undirected graph $G=(V,E)$, nonnegative vertex weights on nonterminal vertices, two protected terminals $s_1,s_2$, and a target terminal $t$. The goal is to delete a minimum-weight set of nonterminal vertices so that $t$ is disconnected from the protected terminals, while $s_1$ and $s_2$ remain connected. This problem captures a basic ``separate while preserve'' requirement that arises in biological intervention design, image analysis with connectivity constraints, and cyber-security attack graph mitigation, where deleting or blocking a node represents preventing the corresponding action, state, or biological entity from participating in a harmful pathway. We prove two results. First, the weighted planar version of three-terminal \RPMNC{} is NP-complete. The reduction is from \textsc{Independent Set} on 3-regular Hamiltonian planar graphs and uses a one-sided blocker construction. Second, we give a polynomial-time $O(\sqrt n)$-approximation algorithm for general graphs. The algorithm is based on an exact path--separator identity, a directed split-graph representation of rooted vertex separators, and a root-linear approximation of a monotone submodular separator function.

Threshold Minimum Cut with Terminal Quotas: Logarithmic and Planar Approximation Algorithms

arXiv: Data Structures and Algorithms — 2026-06-16 00:00:00 UTC

Authors: Qi Duan

We study threshold minimum cut problems with a distinguished root vertex, a set of terminals, and a quota. In the threshold minimum edge cut problem (\TMEC), the goal is to find a minimum-cost edge cut that disconnects at least $k$ terminals from the root. In the threshold minimum node cut problem (\TMNC), the goal is to delete a minimum-cost set of nonterminal, nonroot vertices so that at least $k$ terminals become disconnected from the root. We prove three approximation guarantees. First, undirected general-graph \TMEC{} admits a randomized polynomial-time expected $O(\log n)$ approximation via a Räcke-style cut-dominating tree decomposition and an exact dynamic program on trees. A standard repetition argument gives the same asymptotic ratio with high probability. Second, planar \TMEC{} admits a factor-$2$ approximation by reducing the threshold condition to planar weighted balanced cut. Third, bounded-degree planar \TMNC{} admits a $2Δ$-approximation, where $Δ$ is the maximum degree of a deletable vertex, by reducing the node-cost problem to the planar edge-cut problem on the same graph. The results separate exact-quota guarantees from bicriteria small-set-expansion-type guarantees and identify the unbounded-degree planar node-cut case as the main remaining obstacle.

A constant-factor approximation of the Gromov-Hausdorff distance in the plane

arXiv: Data Structures and Algorithms — 2026-06-16 00:00:00 UTC

Authors: Sushovan Majhi

We give the first polynomial-time constant-factor approximation of the Gromov--Hausdorff distance $d_{GH}$ between finite point sets in the Euclidean plane; in fixed Euclidean dimension such an approximation was previously known only on the line (Majhi, Vitter, and Wenk, 2024). Its engine is the bijective (bottleneck) Gromov--Hausdorff distance $d_{GH}^{bij}$: for two equal-size sets the least additive distortion $\max_{i,j}|d_X(i,j) - d_Y(σi, σj)|$ of a bijection $σ$ equals $2\,d_{GH}^{bij}$, which we likewise approximate within an absolute constant. Approximating additive distortion goes back to Hall and Papadimitriou (2005), who gave a $2$-approximation on the line and observed approximation within $3$ to be NP-hard in dimension three; the planar case they left open is the one we settle. A fat-or-collinear dichotomy drives both bounds: a fat set is aligned by a single rigid motion, while a near-collinear set is split into clusters matched along their dendrogram in one flat, scale-free pass, with relative orientations and per-node reflection signs -- at every scale of the dendrogram -- recovered by global cuts. Relaxing bijections to correspondences yields $d_{GH}$ itself, which reduces to a lone within-cluster-multiplicity kernel -- the pairs an optimal correspondence collapses -- that the same theory closes. Matching lower bounds -- a dimension drop, a multiplicity gap, and a reflection barrier acting at every scale -- show each ingredient is necessary.

Coresets for Continuous $k$-Center in Hyperbolic Space

arXiv: Data Structures and Algorithms — 2026-06-16 00:00:00 UTC

Authors: Eunku Park

We construct coresets for the continuous $k$-center problem in fixed-dimensional hyperbolic space $\mathbb H^D$. The input is a set $P$ of $n$ points in $\mathbb H^D$, where $D=O(1)$, and the centers may be placed anywhere in the ambient hyperbolic space. Given $\varepsilon\in(0,1)$, we construct a subset $P_\varepsilon\subseteq P$ such that every optimal continuous $k$-center solution for $P_\varepsilon$ is a $(1+\varepsilon)$-approximation for $P$. The main difficulty is the exponential volume growth of hyperbolic balls, which prevents a direct grid-based coreset from having size independent of the input radius. We overcome this by dividing the construction according to the farthest-first scale. At bounded scales, we use local Euclidean grids in the Poincaré ball model. At intermediate scales, we use an anchor-centered shell--cone decomposition together with exact distance profiles obtained from the hyperbolic law of cosines. At large scales, we avoid discretizing the ambient ball and instead keep input witnesses indexed by coarse profiles of the induced $k$-center distance functions on each shell--cone bucket. The resulting coreset has size $\left(1/\varepsilon\right)^{O(kD)}$ and can be constructed in time $O(nk\left(1/\varepsilon\right)^{O(kD)}).$ Both bounds are independent of the input radius, and the coreset size is also independent of $n$. Consequently, for fixed $D$, $k$, and $\varepsilon$, this gives a linear-time construction of a constant-size coreset for the continuous $k$-center problem in hyperbolic space.

Online Matching with KIID Edge Arrivals

arXiv: Data Structures and Algorithms — 2026-06-16 00:00:00 UTC

Authors: Yilong Feng, Haolong Li, Xiaowei Wu

In the classic online stochastic matching proposed by Feldman et al. (FOCS 2009), there is a known bipartite type-graph, where one side of the graph is given offline. Upon the arrival of each online vertex, its type is sampled independently and identically from the other side of the type-graph. This model has been extensively studied over the past decade, yielding a rich body of theoretical results. In this paper, we initiate the study of an edge arrival model for online stochastic matching. In our model, the online edges are sampled independently and identically (KIID) from a known type-graph, which need not be bipartite. We first show that the Greedy algorithm cannot achieve a competitive ratio strictly better than $0.5$ while the Suggested Matching algorithm has a competitive ratio of $1-1/e$ under the assumption of integral arrival rates, matching its performance in the one-sided vertex arrival model. We then propose a two-stage algorithm that combines Greedy and Suggested Matching, and show that its competitive ratio is strictly higher than $1-1/e$ for integral arrival rates. While our algorithm is simple, its analysis is intricate and builds upon the Natural LP, which has been proven very powerful in vertex arrival models. Our result reveals that even in the more challenging edge arrival setting for general graphs, competitive ratios better than $1-1/e$ are still possible, given the known distributions.

Single-item lot sizing problem under budgeted lead-time uncertainty

arXiv: Data Structures and Algorithms — 2026-06-16 00:00:00 UTC

Authors: Romain Guillaume, Adam Kasperski, Szymon Wrobel, Pawel Zielinski

In this paper, a single-item lot sizing problem with backordering is discussed. The time horizon is divided into planning periods, characterized by fixed and variable production costs, and future delivery periods with specified demands, where inventory holding and backordering costs may occur. For each planning period, a common nominal lead time is given. The true lead times can deviate to some extent from the nominal one, and their exact values are unknown at the planning step. We assume that lead times take only integer values and splitting production orders is not allowed. Furthermore, order crossovers are prohibited; thus, an order placed earlier cannot arrive after one placed later. A budgeted uncertainty set of possible lead-time scenarios is defined, where a budget allows us to control the amount of uncertainty of lead times. It is shown how to construct a family of production plans varying from the most optimistic (a best lead-time scenario occurs) to the most pessimistic (a worst lead-time scenario occurs). In order to compute these plans the R* criterion is applied which generalizes the conservative robust min-max criterion, commonly used in robust optimization. The computational complexity of the problem is investigated. Polynomial, pseudopolynomial time algorithms, and mixed integer programming formulations are proposed to solve the general problem and its special cases. The results of computational tests are provided that demonstrate that using the R* criterion can significantly enlarge the set of candidate production plans.

C^2: Cache-Conscious Succinct Tries with Adaptive Unary Path Compression

arXiv: Data Structures and Algorithms — 2026-06-16 00:00:00 UTC

Authors: Kepan Zhang, Tiancheng Zhao, Helen Xu

Succinct tries are powerful string dictionaries because of their low memory footprint and fast query performance. However, existing succinct trie implementations face two key challenges to spatial locality: 1) they incur unnecessary cache misses during queries, especially during trie navigation operations, and 2) they waste significant space when the data contains many unary paths. We propose C^2, a set of two techniques: C_1 introduces a more cache-friendly layout for the \bv underlying succinct tries, and C_2 compresses redundant unary paths. We thoroughly redesign three state-of-the-art succinct tries: FST, CoCo-trie, and Marisa, producing C^2-FST, C^2-CoCo, and C^2-Marisa. Experiments on six diverse datasets show that the C_1 optimization improves query performance by 1.58x, 1.12x, and 1.42x, respectively, compared to the original FST, CoCo-trie, and Marisa. Furthermore, the C_2 optimization achieves a 1.3x smaller memory footprint on average. The succinct tries optimized with both aspects of C^2 achieve better space-time tradeoffs than their original versions and other state-of-the-art succinct tries, while using significantly less space than non-succinct tries like ART and C-ART.

Contested Cluster Selectors: Local Ambiguity, Normal Forms, and Backtracking Cost in Random Constraint Satisfaction

arXiv: Data Structures and Algorithms — 2026-06-16 00:00:00 UTC

Authors: Karthik Sheshadri

We introduce and empirically investigate \emph{contested cluster selectors} (\CCS): variables that are non-backbone, carry information about solution-cluster identity, and are repeatedly but unreliably forced by local propagation during backtracking search. In instrumented \DPLL{} experiments on random 3-\SAT{} near the empirical satisfiability threshold and on near-optimal random \VC{} instances, a small number of such variables accounts for a large fraction of observed backtracking cost. Pinning two or three high-contestedness variables to solution-consistent values reduces backtracking by 70--80\% on the reference instances studied, and a static degree--polarity metric yields a simple $2^k$ enumeration heuristic with a reported $3.7\times$ speedup over baseline \DPLL{} at $n=50$. A polynomial control experiment on random 3-\XORSAT{} sharpens the interpretation. Gaussian elimination exposes the true affine selector coordinates, whereas \DPLL{} churn concentrates on pivot variables chosen in a poor coordinate system. Thus clustering and non-backbone status are not enough: the empirical hardness signal is \emph{local contestation} that remains after available polynomial-time normal forms. We formalize this distinction through safe coordinate exposers and the \emph{unavoidable contested selector cost} (\UCSC). We also prove an ordered single-pass eraser-memory lower bound: any ordered \FERAM{} that recovers a $k$-bit cluster label from a distribution with residual min-entropy $k-η$ using $S$ bits succeeds with probability at most $2^{S+η-k}$. The paper positions \CCS/\UCSC{} as a structural program connecting backdoors, solution-space geometry, low-degree barriers, and Schaefer-style algebraic normal forms. We do not claim a proof of $P\ne NP$; rather, we isolate the normal-form barrier that any such extension would need to overcome.

Active Learning with Low-Rank Structure for Data Selection

arXiv: Data Structures and Algorithms — 2026-06-16 00:00:00 UTC

Authors: Vincent Cohen-Addad, Sasidhar Kunapuli, Vahab Mirrokni, Mahdi Nikdan, David P. Woodruff, Samson Zhou

In the data selection problem, the objective is to choose a small, representative subset of data that can be used to efficiently train a machine learning model. Sener and Savarese [ICLR 2018] showed that, given an embedding representation of the data and suitable geometric assumptions, heuristics based on $k$-center clustering can be used to perform data selection. This perspective was further explored by Axiotis et. al. [ICML 2024], who proposed a data selection approach based on $k$-means clustering and sensitivity sampling. However, these methods rely on the assumption that the dataset exhibits intrinsic geometric structure that can be effectively captured by clustering, whereas many modern datasets instead possess global algebraic structure that is better exploited by low-rank approximation or principal component analysis. In this paper, we introduce a new data selection framework based on low-rank approximation and residual-based sampling, formulated through the lens of row subset selection and loss-preserving coreset construction. Given an embedding representation of the data satisfying mild regularity conditions, which can be interpreted as algebraic or angular notions of Lipschitz continuity, we show that it is possible to select a weighted subset of $\tilde{O}\left(k + \frac{1}{\varepsilon^2}\right)$ data points whose average loss approximates the average loss over the full dataset within a $(1+\varepsilon)$ relative error, up to an additive $\varepsilon Φ_k$ term, where $Φ_k$ denotes the optimal rank-$k$ approximation cost of the embedding matrix. We complement these theoretical guarantees with empirical evaluations, demonstrating that on a range of real-world datasets, our data selection approach achieves improved performance over prior strategies based on uniform sampling or clustering-based sensitivity sampling.

Raiders of the Lost Log: Synchronous Parallel In-Place Models and Algorithms

arXiv: Data Structures and Algorithms — 2026-06-16 00:00:00 UTC

Authors: Michael T. Goodrich, Vinesh Sridhar

Embedded systems and Internet of Things (IoT) applications motivate in-place parallel algorithms, which avoid allocating additional shared memory past the input. Work by Gu, Obeya, and Shun [APOCS '21] defines a family of PIP (parallel in-place) models and parallel algorithms that eschew auxiliary memory at high processor counts while remaining in-situ when run sequentially. However, their models assume asynchronous processing and have no in-place guarantees for intermediate processor counts. We address this gap in the literature by proposing a Synchronous PIP family of models for in-place parallel and distributed computation. We demonstrate the effectiveness of our new model by giving efficient and synchronous parallel algorithms in this model that require no auxiliary shared memory and only constant private memory per processor. Importantly, we show how to leverage a new parallel-augmented sweep technique to ensure that Synchronous PIP algorithms remain efficient and strictly in-place at all processor counts.

Resizable Retrieval

arXiv: Data Structures and Algorithms — 2026-06-16 00:00:00 UTC

Authors: William Kuszmaul, Aaron Putterman, Tingqiang Xu, Hangrui Zhou, Renfei Zhou

A dynamic retrieval data structure encodes a function $f:K \rightarrow [2^v]$ for a set $K \subseteq [U]$, while supporting queries $f(x)$ for $x\in K$, insertions \texttt{Insert}$(x, f(x))$ for $x \notin K$, and deletions \texttt{Delete}$(x)$ for $x \in K$. Given an upper bound $N$ on $|K|$, it is known how to solve the dynamic retrieval problem with $O(1)$-time operations and space $Nv + O(N \log \log (U/N))$ bits. An open question, first posed by Demaine et al. in 2006, is whether a similar bound can be achieved with a resizable data structure, whose space bound is parameterized by the \emph{current} size $n$ of $K$. We answer this question in the affirmative and prove matching lower bounds for the space-time trade-off achieved by our data structure. We also give corollaries for space-efficient memory allocation and dynamic filters.

Modern Primal-Dual Frameworks for Prior-Free Online Resource Allocation

arXiv: Data Structures and Algorithms — 2026-06-16 00:00:00 UTC

Authors: Rad Niazadeh, Rajan Udwani

Linear-programming (LP)-based primal-dual methods are fundamental for designing and analyzing algorithms in adversarial (prior-free) online resource allocation. This chapter provides a tutorial on two modern primal-dual frameworks, emphasizing recent developments and contemporary models in operations research. Part~I develops an LP-based convex-programming framework where solving a regularized convex program at each arrival captures the tradeoff between greediness and hedging, yielding a dual certificate via Karush-Kuhn-Tucker (KKT) conditions. Because standard LP relaxations can be weak or intractable for stochastic outcomes, Part~II introduces a complementary LP-free framework that provides a universal certificate system for evaluating competitive ratios under such uncertainty. Covering a wide array of models -- including online vertex-weighted bipartite matching, edge-weighted online matching with free disposal, online matching with stochastic rewards, reusable resources, two-sided assortment optimization, configuration allocation (whole-page optimization), AdWords, and costly cancellations -- the tutorial equips readers with versatile proof templates to analyze existing algorithms and develop new solutions for emerging applications.

Distributed Dominating Set With Optimal Rounds and Message Size in Bounded Arboricity Graphs

arXiv: Data Structures and Algorithms — 2026-06-16 00:00:00 UTC

Authors: Sharareh Alipour, Ermiya Farokhnejad

We study the distributed minimum dominating set problem on graphs of arboricity $α$. Dory, Ghaffari, and Ilchi [PODC'22] showed that any algorithm achieving a constant or poly-logarithmic approximation factor needs at least $Ω(\logΔ/\log\logΔ)$ rounds in graphs of maximum degree $Δ$ and arboricity $α$, even when $α=2$ and even when the message sizes are unbounded. Although there is a variety of algorithms with a near-optimal round complexity of $O(\logΔ)$, it is natural to ask: What is the best approximation factor in the optimal round complexity of $O(\logΔ/\log\logΔ)$? We make progress in answering this question by describing a deterministic algorithm that obtains a $O\left( α\log Δ/ \log\log Δ\right)$ approximation without prior knowledge of $α$ with optimal round complexity of $O\left( \log Δ/ \log\log Δ\right)$ and optimal message size of $1$ bit per round. Among all of the previous results, the only algorithm that achieves the optimal round complexity of $O\left( \log Δ/ \log\log Δ\right)$ without prior knowledge of $α$ is due to Lenzen and Wattenhofer [DISC'10] that obtains a $O(α\log^{1+\varepsilon}Δ/ (\varepsilon\log\log Δ))$ approximation in $O(\logΔ/(\varepsilon\log\logΔ))$ rounds and $O(\log(\varepsilon^{-1}\logΔ))$ message size. Our algorithm simplifies and improves upon this result. The only downside of our algorithm compared to the algorithm of Lenzen and Wattenhofer is that it needs prior knowledge of $Δ$. The previous state-of-the-art algorithm by Dory, Ghaffari, and Ilchi [PODC'22] has a dependency on $\log n$ in the round complexity for unknown $α$, which is far from optimal.

Linear algebra at exponential scale via tensor network dimension reduction

arXiv: Data Structures and Algorithms — 2026-06-16 00:00:00 UTC

Authors: Chris Camaño, Ethan N. Epperly, Raphael A. Meyer, Joel A. Tropp

Many problems in modern scientific computing are challenging because of a \emph{curse of dimension}, where their mathematical formulation involves objects whose dimension is \emph{exponential} in the nominal "size" of the problem. Tensor networks can provide a compact representation for exponentially large vectors and matrices that arise in applications, but these representations do not always lead to reliable algorithms. This paper develops and analyzes techniques for randomized dimension reduction of tensor network data. These techniques support a suite of efficient algorithms for provably solving exponential-scale linear algebra problems, including trace estimation and eigenvalue approximation. The paper includes several stylized illustrations from quantum many-body physics with ambient dimension up to $2^{200}$.

Can Neural Networks Achieve Optimal Computational-statistical Tradeoff? An Analysis on Single-Index Model

arXiv: Data Structures and Algorithms — 2026-06-16 00:00:00 UTC

Authors: Siyu Chen, Beining Wu, Miao Lu, Zhuoran Yang, Tianhao Wang

In this work, we tackle the following question: Can neural networks trained with gradient-based methods achieve the optimal computational-statistical tradeoff in learning Gaussian single-index models? Prior research has shown that any polynomial-time algorithm under the statistical query (SQ) framework requires $Ω(d^{s^\star/2}\lor d)$ samples, where $s^\star$ is the generative exponent representing the intrinsic difficulty of learning the underlying model. However, it remains unknown whether neural networks can achieve this sample complexity. Inspired by prior techniques such as label transformation and landscape smoothing for learning single-index models, we propose a unified gradient-based algorithm for training a two-layer neural network in polynomial time. Our method is adaptable to a variety of loss and activation functions, covering a broad class of existing approaches. We show that our algorithm learns a feature representation that strongly aligns with the unknown signal $θ^\star$, with sample complexity $\widetilde{O} (d^{s^\star/2} \lor d)$, matching the SQ lower bound up to a polylogarithmic factor for all generative exponents $s^\star\geq 1$. Furthermore, we extend our approach to the setting where $θ^\star$ is $k$-sparse for $k = o(\sqrt{d})$ by introducing a novel weight perturbation technique that leverages the sparsity structure. We derive a corresponding SQ lower bound of order $\widetildeΩ(k^{s^\star})$, matched by our method up to a polylogarithmic factor. Our framework, especially the weight perturbation technique, is of independent interest, and suggests potential gradient-based solutions to other problems such as sparse tensor PCA.

Comparison Patrols on Drifting Orders: Certified Rank Maintenance, Evolving Planar Maxima, and Selection under Drifting Fitness

arXiv: Data Structures and Algorithms — 2026-06-16 00:00:00 UTC

Authors: Faruk Alpay, Levent Sarioglu

Rank-based selection in dynamic environments acts on order information that becomes stale while it is being used. Tournaments, elitism, truncation, and Pareto selection may therefore consume rankings that no longer match the current fitness order, while full re-evaluation competes with search for the same budget. This paper formulates the missing information layer as a data-structure problem. A hidden total order on $n$ items drifts by adjacent transpositions, while a maintainer receives one truthful pairwise comparison per step and must answer rank queries continuously. We introduce the comparison patrol, a constant-time maintained-order structure using $3n+O(1)$ words, one comparison per update, deterministic verification-age bounds, and per-item displacement certificates. We prove lower bounds showing that oblivious and location-oblivious maintainers incur expected Kendall error $Ω(\min(α,1)n)$, and show that the patrol operates at the same order. A bump invariant yields exact self-stabilization after drift-free corruption: if the maximum rank overstatement is $L$, recovery takes at most $L$ aligned cycles and cannot finish before $L-1$. This gives a deterministic shock-recovery calculus and a crossover with full rebuild near $L\approx \log_2 n$. The maintained order is then transferred to evolving planar maxima and to evolutionary selection rules, giving deterministic bounds for truncation, tournament, elitist, and two-objective Pareto decisions under drifting fitness. Experiments up to $n=65{,}536$ audit the certificates, recovery laws, equilibrium behavior, and equal-budget dynamic evolutionary loops, identifying when certified local rank maintenance outperforms global re-evaluation and when it should hand over.

Optimality of Random Regular Graphs in Sparse Network Designs

arXiv: Data Structures and Algorithms — 2026-06-16 00:00:00 UTC

Authors: Weijia Li, Xiaochun Niu, Yehua Wei, Jiaming Xu

The problems of designing sparse networks arise frequently in resource allocation and operations research. In production systems, for example, sparse process flexibility designs are used to handle uncertain demand effectively: the goal is to construct the sparsest bipartite graph between supply and demand that still achieves an expected fulfilled demand comparable to that of a fully flexible system. In middle-mile transportation, sparse delivery-route subgraphs that sustain large matchings after random node deletions help reduce delivery costs; here, the goal is to design the sparsest graph whose maximum matching size remains comparable to that of the fully connected graph under node deletions. The design of sparse networks has been studied extensively, with state-of-the-art results providing order-wise optimal designs for both bipartite and unipartite networks (Chen et al., 2015; Feng et al., 2024). However, identifying designs that achieve the sharp theoretical limit -- where the average degree asymptotically matches the lower bound of any graph to achieve a given loss level, has remained open. In this paper, we prove that the random regular graph achieves this sharp optimal condition in both bipartite and unipartite settings. Numerical experiments further validate this optimality. Our results highlight a practical guideline for sparse flexibility networks: designs that combine degree regularity with low edge correlations can achieve optimal performance under uncertainty.

Differentially Private Submodular Maximization with a Knapsack Constraint

arXiv: Data Structures and Algorithms — 2026-06-16 00:00:00 UTC

Authors: Ron Zadicario, Tova Milo

Submodular maximization subject to a knapsack constraint (SMK) is a fundamental problem in discrete optimization, with wide-ranging applications in machine learning and related fields. As these applications increasingly involve sensitive individual data, there is a growing need for high-utility algorithms that provide formal privacy guarantees. In this work, we study the SMK problem under differential privacy, considering both monotone and non-monotone objective functions. For monotone objectives, we propose a differentially private algorithm that achieves the optimal $(1-1/e)$-approximation ratio while significantly improving both additive error and query complexity over prior work. We also present a more efficient algorithm for the same setting, achieving a $1/2$-approximation. For non-monotone objectives, we introduce, to our knowledge, the first differentially private algorithm with provable guarantees, achieving a $1/4$-approximation in expectation and an additive error comparable to the best known for monotone objective functions.

Linkage

David Eppstein — 2026-06-15 16:56:00 UTC

Biomedical papers with fabricated references blew up by more than a factor of ten from 2023 to 2026 ($\mathbb{M}$, via). Relatedly, arXiv has added a new policy banning author of papers with hallucinated citations or other artifacts of unchecked LLM use for a year (and after that permanently requiring publication prior to allowing a preprint, kind of problematic for publication in journals whose submission process depends on having an arXiv preprint. Unfortunately I can’t find a direct reference to this in the actual posted arXiv policies and I’m not going to link to X/twitter where Tom Dietterich posted the only semi-official communication I can find on it.
Fish curve drawn as an epicycle. See also “Wheels on wheels on wheels” by Frank Farris in Math. Mag. on symmetric doily patterns drawn using epicycles.
I recently learned that the cubic and quartic cases of Fermat’s last theorem (the impossibility of integer solution of $x^3+y^3=z^3$ and $x^4+y^4=z^4$) go back farther than Fermat ($\mathbb{M}$), to medieval Islamic mathematics. Abu-Mahmud Khujandi tried to prove the impossibility of the cubic case (but the proof was faulty) and later Ibn al-Khawwām included both cases in a list of open problems.
The age of an open problem is not a good proxy for its difficulty or importance, notes Lance Fortnow.
Another day, another journal editorial board mass resignation ($\mathbb{M}$, via) this time from Springer’s Natural Language Semantics, over publisher demands to increase article publication charge revenue by increasing their number of acceptances. Instead the same editors are launching a new diamond open-access journal, Semantics of Natural Languages, through the Open Library of Humanities.
The American Diabetes Association has ejected from its annual meeting the editor-in-chief of its journal Diabetes Care, its former president, and three other scientists ($\mathbb{M}$), for handing out reprints of an editorial published in the journal and critical of the US government’s attacks on science. According to video evidence the ejected scientists “were not disruptive or disorderly in their conduct”. Nevertheless, the ADA claims that this behavior constituted a violation of the association’s code of conduct.
On circle packing with more than one size of circle.
Jeremy Kun (1) enjoys a rambly substack post on software developer bros who don’t even know how to find the center of a triangle, and how this somehow portends the obsolescence of software development (2) excuses the bros from their ignorance on the basis that there are “at least 26 different definitions of a triangle’s center”, (3) discovers that the Encyclopedia of Triangle Centers has “actually over 71 thousand definitions”, (4) realizes that his own ignorance has made him become the software bro he was enjoying the substack diss of, and (5) describes the whole saga, as if waking from a dream.
Terry Tao clearly explains why proposed new US funding rules requiring conference travel to be planned years ahead of time as part of grant proposals do not work for the time scales of mathematics research ($\mathbb{M}$). These rules would be even more disastrous for computer science, where conference publication is essential for making one’s research known and where one cannot know which conferences the research will appear in until the conference acceptance and rejections come in, only a few weeks or months before the conference itself.
How to wrap a hat tiling around a cylinder such as, say, a coffee mug, if you had a manufacturing process for them that didn’t involve leaving a seam in the design.
New preprint: “Existence of a periodic orbit for billiards in polygons ($\mathbb{M}$), Giovanni Forni. I think the title is self-explanatory; this claims to solve a major open problem in geometric dynamical systems. (I have too little expertise in this topic to evaluate its correctness.)
My 2024 paper “Non-crossing Hamiltonian paths and cycles in output-polynomial time](https://ics.uci.edu/~eppstein/pubs/p-noncross.html)” has been given the inaugural 2025 Algorithmica Best Paper Award ($\mathbb{M}$) as the most impactful paper in the journal Algorithmica from the past three years. The paper shows that one can list all of the traveling salesperson tours through a set of points in the plane in an amount of time polynomial in the number of tours. This number can be exponential, but the harder part of the problem is finding the tours quickly when there are fewer than exponentially many of them.
Dissident Russian mathematician Mikhail Verbitsky, traveling from Brazil to an award ceremony in Armenia, was arrested there at the request of the Russian government ($\mathbb{M}$). The linked article, in French, is from the French Mathematical Society, which strongly protests this action. As of posting he is said to be freed on a technicality but forbidden from leaving Armenia.
Seventeen camels and where they can take you ($\mathbb{M}$). Jim Propp on how “six seemingly unrelated brain-teasers” including the 17-camel inheritance problem as well as exercises on counting edges in forests can be solved by the same trick for dealing with linear equations with constant offsets.

By David Eppstein

TR26-101 | On Proof Systems for #QBF | Sravanthi Chede, Leroy Chew, Vaibhav Krishan, Anil Shukla

ECCC Papers — 2026-06-15 08:11:15 UTC

For a quantified Boolean formula (QBF), the problem of computing the number of winning strategies is known as the #QBF problem. This problem is considered harder than the analogous #SAT problem. Recently, important proof systems for QBFs and #SAT have been studied. By extending the ideas from both fields, we show that it is possible to design proof systems for #QBF. Such proof systems are important not only for advancing the theory of #QBF but also for certifying and designing better #QBF solvers, an area that is still in its early stages. In this paper, we explore #QBF proof systems to count the number of Skolem functions. In addition to a naive system, we study #QBF systems based on the $\forall$-expansion rule of QBFs. We observe that these systems have inherent structural weaknesses that lead to lower bounds. As an alternative, we propose a #QBF proof system that we call Q-MICE, which consists of sound inference rules for computing and certifying the #QBF solution, similar to the line-based #SAT proof system MICE. To demonstrate the strength of Q-MICE, we present various upper bounds, such as the quantified version of the propositional XOR-PAIRS formula, which is known to be hard for MICE. Consequently, we also separate Q-MICE from $\forall$-expansion based #QBF proof systems.

Theory of Computing Report

Call for workshop proposals: FOCS 2026

On the Complexity of the Circuit Width Problem

Blended Chart Surfaces: A Seamless Explicit Representation for Smooth Surface Fitting

Greedy Vector Balancing

High-Fidelity 3D Geometric Reconstruction of Pelvic Organs from MRI: A Hybrid Deep Learning and Iterative Optimization Approach

Non-negative Matrix Factorisation with Topological Regularisation

Agent Utilities over Generalized Voronoi Regions and their Gradients

Directed Reachability-Preserving Minimum Edge Cut: Approximation and Planar Hardness

A Counterexample to Wegner's Conjecture for Axis-Parallel Rectangles

Online Connectivity Augmentation

Reducing Prize-Collecting Stroll and Related Routing Problems to Prize-Collecting TSP

A polynomial-time approximation scheme for minimum-weight decoding of topological codes

The Value of Adaptivity in LSM Bloom-Filter Tuning: A Log-Law and a Two-Clock Frontier

The independence number of uncrowded hypergraphs: bounds matching the shattering threshold

Four-Cycle Counting in Low-Degeneracy Graph Streams

The 2026 Algorithmic Information Theory Data Compression Challenge

Spectral recovery of a planted triangle-dense subgraph

Exact Algorithms for Edge Deletion to Cactus Graphs and Weighted Variants

Approximation Preserving Coresets

Space-Efficient Lock-Free Linear-Probing Hash Table

Scalable K-clique Estimation with Differential Privacy

Adaptive Proximal Methods for Weakly Convex Optimization with Unknown Parameter: Deterministic and Stochastic Guarantees

Sum-of-Squares Degree Barriers for the Reweighted-Hinge Method in Robust Halfspace Learning: A Christoffel-Function Characterization

The Complexity of Min-Max Optimization for Quadratic Polynomials

Worst-case depth hierarchy for shallow quantum circuits

The Computational Complexity of Team Zero-Sum Games

Revisiting average case complexity of multilevel syllogistic: From the 1995 Courant Technical Report to Lean 4 Formalization

Polynomial-Time Mistake-Bounded Language Generation

Quiet Planting for $k$-SAT, Multiple Solutions of Arbitrary Geometry

The Exact Reach of Conormal Invariants in Determinantal Complexity: a Quadratic No-Go Theorem

Three-Terminal Reachability-Preserving Minimum Node Cut: Planar Hardness and a General-Graph \(O(\sqrt n)\)-Approximation

Threshold Minimum Cut with Terminal Quotas: Logarithmic and Planar Approximation Algorithms

A constant-factor approximation of the Gromov-Hausdorff distance in the plane

Coresets for Continuous $k$-Center in Hyperbolic Space

Online Matching with KIID Edge Arrivals

Single-item lot sizing problem under budgeted lead-time uncertainty

C^2: Cache-Conscious Succinct Tries with Adaptive Unary Path Compression

Contested Cluster Selectors: Local Ambiguity, Normal Forms, and Backtracking Cost in Random Constraint Satisfaction

Active Learning with Low-Rank Structure for Data Selection

Raiders of the Lost Log: Synchronous Parallel In-Place Models and Algorithms

Resizable Retrieval

Modern Primal-Dual Frameworks for Prior-Free Online Resource Allocation

Distributed Dominating Set With Optimal Rounds and Message Size in Bounded Arboricity Graphs

Linear algebra at exponential scale via tensor network dimension reduction

Can Neural Networks Achieve Optimal Computational-statistical Tradeoff? An Analysis on Single-Index Model

Comparison Patrols on Drifting Orders: Certified Rank Maintenance, Evolving Planar Maxima, and Selection under Drifting Fitness

Optimality of Random Regular Graphs in Sparse Network Designs

Differentially Private Submodular Maximization with a Knapsack Constraint

Linkage

TR26-101 | On Proof Systems for #QBF | Sravanthi Chede, Leroy Chew, Vaibhav Krishan, Anil Shukla