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cs.DS: explore 187 source-linked works published from 2022 to 2026, with original documents and citations.

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Includes records with this source-supplied label or an explicit phrase match in their metadata. Matches indicate a mention, not proof that a paper uses a method or tests a material. Source versions are consolidated by DOI.

Sources: arxiv. Collection updated 2026-09-16. Counts describe this index, not the complete source archives.

arXiv2026

On Maximizing a Weakly Submodular Function over a Matroid Constraint via the Greedy Algorithm

We consider the problem of approximately maximizing a weakly submodular function using the standard greedy algorithm, which is known to give tight approximation results for such functions under a cardinality constraint. We show that this is not the case for general matroid constraints. For any $γ< 1$, we give a family of $γ$-weakly submodular functions and a simple partition matroid constraint and show that the standard greedy algorithm provides no constant approximation for the resulting constrained maximization problem.

cs.DS↗
arXiv2026

Algebraic Geometry Codes Approach the Half-Singleton Bound with Constant Field Size

We study linear codes for insertion and deletion (insdel) errors through the lens of evaluation codes. We develop a general framework for analyzing random puncturings of evaluation codes, where the edit distance is controlled by only the size of the evaluation domain and the maximum number of zeros of a nonzero function in the underlying function space. Our proof generalizes the results of Con, Guo, Li, and Zhang (ICALP 2025), and simultaneously simplifies their arguments by avoiding an in-depth analysis of longest common subsequences. We demonstrate the applicability of our core theorem by instantiating it with random puncturings of Reed--Muller codes. We then recover the result that random Reed--Solomon codes approach the half-Singleton bound over linear-sized fields while also improving the dependence on the additive gap $\varepsilon$ from $2^{O(1/\varepsilon^2)}$ to $2^{O(1/\varepsilon)}$. Finally, by applying the framework to algebraic geometry codes arising from asymptotically good towers of function fields, we show that there exist randomized families of structured linear codes over constant-sized fields that approach the half-Singleton bound.

cs.IT↗
arXiv2026

Online Matching in Convex Bipartite Graphs

Online resource-allocation systems, like outpatient scheduling and spectrum allocation, often assign sequentially arriving requests to an ordered pool of scarce resources, where each request accepts a contiguous interval of feasible options. We study the resulting online matching problem on convex bipartite graphs under irrevocable decisions and adversarial arrivals. We first show that convexity alone does not improve the classic worst-case guarantee of 1-1/e, achieved by Ranking. We then consider the uniform-length model, in which every online request has exactly d consecutive offline neighbors. We propose Flip, which uses one random bit to commit ex-ante to either earliest-feasible assignment or latest-feasible assignment. Although either natural deterministic policy can waste capacity and be asymptotically only 1/2-competitive, we show that their randomized mixture is 2/3-competitive. This guarantee is tight for Flip and remains valid against a semi-adaptive adversary that observes the selected policy before choosing the arrival order. We also prove that no randomized online algorithm can achieve a competitive ratio strictly larger than 3/4 in the uniform-length model.

cs.DS↗
arXiv2026

Strategic Facility Location in Euclidean Spaces

The strategic facility location problem is defined as follows: $n$ agents report their location in a metric space, and the objective is to design a \emph{mechanism} deciding the (possibly randomized) location of a facility such that agents have no incentive to lie about their position. We focus on the egalitarian cost, which means that the goal of the mechanism is to minimize the expected maximal facility-agent distance. Meanwhile, mechanisms must be \emph{truthful} (or \emph{strategyproof}): no agent may decrease their expected distance to the facility via lying on their location. Designing truthful mechanisms minimizing the approximation ratio is a well-studied problem, and the optimal solution is known for the real line. We focus in this paper on higher dimension Euclidean spaces, for which gaps remain between the best known lower and upper bounds. We first show that, maybe counter-intuitively, the problem is easier for two agents on the plane rather than on the line: the mechanism can exploit the additional dimension to prevent more efficiently agent lies. Based on this intuition, we devise lower bounds for $\mathbb R^d$ asymptotically matching the best known approximation factor of $2$ for large $d$. We also provide novel mechanism ideas, improving over the best known algorithms on the plane, and when the agents belong to $\mathbb R^d$ but the facility may use an additional dimension.

cs.GT↗
arXiv2026

Machine Unlearning as Private Retroactive Algorithms

Machine unlearning typically aims to emulate retraining from scratch: upon a deletion request, the unlearning algorithm should produce an outcome that would have been obtained had the deleted point never been included. Recent work has shown that this emulation requirement carries no meaningful privacy semantics against an adversary who observes a sequence of releases. Machine unlearning is thus not a privacy question per se, but rather a data maintenance question, which is precisely the subject of retroactive algorithms. These are algorithms supporting modifications of past operations, guaranteeing that all subsequent answers reflect the revised history as if it had always been in force. We put forward a definition of private retroactive algorithms, combining the retroactivity requirement with differential privacy under continual observation. We present constructions achieving both privacy and retroactivity at no asymptotic cost over privacy alone for linear statistics, clustering, and histograms, alongside impossibility results.

cs.CR↗
arXiv2026

Integrality Gap Bounds for the Goemans-Linial SDP on Finite Abelian Cayley Graphs

In the uniform sparsest cut problem we are asked to find a vertex set that cuts few edges relative to the number of vertex pairs it separates. The Goemans-Linial SDP coupled with the Arora-Rao-Vazirani rounding gives an $\mathcal{O}(\sqrt{\log n})$ approximation on arbitrary graphs on $n$ vertices. We study this relaxation on finite Abelian Cayley graphs. First we show that when the second normalized Laplacian eigenvalue of $G= \mathrm{Cayley}(Γ, S)$ is realized by a Fourier character with image size at most four then $λ_2(G)=\mathrm{SDP}_{\mathrm{GL}}(G)=ψ(G)$. Geometrically, a character maps the vertices onto a regular polygon where the squared chord distance satisfies the triangle inequalities exactly when the polygon has at most four vertices. Grouping equal character fibers gives a cyclic quotient where the optimal cut can be found exactly and so the relaxation is exact on finite Abelian Cayley graphs on groups of exponent at most four. Second, we replace each generator $s$ of $S$ by a uniformly random element of its cyclic subgroup (including identity). If $r_s$ is the order of $s$, we let $α(r_s)$ to be the average number of $\pm s$ steps needed to simulate such a move, and let $ρ(S)=\max_{s\in S}α(r_s)$ be its worst case. Full cyclic averaging eliminates character phases and choosing a nontrivial character $χ^*$ minimizing the auxiliary eigenvalue and taking $K=\mathrm{ker}χ^*$ gives \[ ψ(G)\leqψ_G(K)\leq\frac{q^*}{q^*-1} \cdotρ(S)\cdot\mathrm{SDP}_{\mathrm{GL}}(G)\leq 2ρ(S)\cdot\mathrm{SDP}_{\mathrm{GL}}(G), \] where $q^*=|χ^*(Γ)|$. If all generator orders are at most $R$, this is an $R/2$ approximation. Finally, we construct an infinite family of finite Abelian Cayley graphs with Goemans-Linial integrality gap exactly $16/15$.

math.CO↗
arXiv2026

Quantum lower bounds for convex optimization and real matrix-vector query problems

We (the author and the AI systems that did the heavy lifting) show that the quantum query complexity of minimizing a convex function over a convex subset of $\mathbb{R}^n$ with evaluation and membership queries is $\tildeΩ(n)$, nearly matching the best known upper bound. In particular, we show this even for quadratic minimization, which is equivalent to inverting an $n \times n$ real matrix using matrix-vector queries. We also show linear or nearly linear lower bounds on the quantum query complexity of computing the trace, the sign of the determinant, and the magnitude of the determinant of a real matrix in the matrix-vector query model. We use a novel quantum lower bound technique, the determinantal witness method, based on identifying a witness whose Fourier transform vanishes on low-rank matrices and that correlates well with the function being computed.

quant-ph↗
arXiv2026

Approximation algorithms for the square min-sum bin packing problem

In this work, we study the Square Min-Sum Bin Packing Problem (SMSBPP), where a list of $n$ square items has to be packed into square bins of dimensions $1 \times 1$ with no overlap between the areas of the items. The bins are indexed (starting at one) and the cost of packing each item is equal to the index of the bin in which it is placed. The objective is to minimize the total cost of packing all items, which is equivalent to minimizing the average cost of items. The problem has applications in minimizing the average time of logistic operations such as cutting stock and delivery of products. We prove that classic algorithms for two-dimensional bin packing that order items in non-increasing order of size, such as Next Fit Decreasing Height or Any Fit Decreasing Height heuristics, can have an arbitrarily bad performance for SMSBPP. We, then, present an algorithm with an approximation ratio of $\sqrt{205}-12+δ$ ($\approx 2.3178+δ$), for any $δ> 0$, and running time $O(n \log n)$. Finally, we also present a PTAS for the problem.

cs.DS↗
arXiv2026

Weighted Emulators with Local Heaviest Edges Stretch for Undirected Graphs

We introduce a generalized family of $\left( 2\cdot \left\lfloor \frac{k}{2} \right\rfloor-1, 2\cdot \left\lceil \frac{k}{2} \right\rceil \cdot W_{1} +\max\left\{0,2\cdot\left(\left\lceil\frac{k}{2}\right\rceil-2\right)\right\}\cdot W_{2} \right)$-emulators with $\tilde O \left(n^{1+\frac{1}{k}}\right)$ edges, for any $k\in\mathbb{N}$, where $W_{i}$ is the $i$th heaviest edge on a shortest path between two vertices. Our construction generalizes the $+2W_{1}$-spanner of size $\tilde O\left(n^{\frac{3}{2}}\right)$ and the $+4W_{1}$-emulator of size $\tilde O \left(n^{\frac{4}{3}}\right)$, both by Elkin, Gitlitz and Neiman [DISC'21 and DICO'23]. When $k$ is even, these are $\left(k-1,k\cdot W_{1} + \left(k-4\right)\cdot W_{2}\right)$-emulators and when $k$ is odd, these are $\left(k-2,\left(k+1\right)\cdot W_{1} + \left(k-3\right) \cdot W_{2}\right)$-emulators. Our framework not only expands known constructions for weighted graphs but also yields an improved stretch over state of the art emulators and spanners for unweighted graphs within a specific distance regime. In particular, for all vertex pairs separated by a distance of $δ\leq O\left(3^{k^{2}}\right)$, our construction improves upon the seminal additive $+\tilde O\left(δ^{1-\frac{1}{k}}\right)$-emulator of size $\tilde O\left(n^{1+\frac{1}{2^{k+1}-1}}\right)$ by Thorup and Zwick [SODA'06].

cs.DS↗
arXiv2026

Assortment and Procurement Design in Dual-Mode Content Platforms

We study assortment and procurement design for a digital content platform offering both ad-supported and subscription access. Users are heterogeneous in content preferences and ad tolerance and self-select between the two modes or an outside option. For a fixed common subscription price and ad load, the platform chooses assortment distributions specific to each user type and access mode, together with content-family-level buy-versus-rent decisions to maximize profit. Rental costs scale with realized consumption, whereas buying provides a reusable pool of titles whose cost depends on the largest induced requirement across user types and modes. We show that the resulting problem is NP-hard. We then develop a scalable approximation framework based on a candidate buy set, a relaxation of the procurement coupling, and a decomposition into linear programs with a single equality constraint. These subproblems are solved by dual bisection with cardinality-constrained assortment optimization, followed by restricted-master postprocessing to recover primal feasibility. The method yields computable optimality-gap bounds, an interpretable threshold-based procurement heuristic, and asymptotic optimality under proportional market scaling as market size and grid resolution increase. Numerical experiments show strong performance at moderate market scales and grid sizes.

cs.DS↗
arXiv2026

Counterfactual Routing Using Integer Programming with Constraint Generation

We present our submission to the IJCAI 2025 'Counterfactual Routing Competition' (CRC 25). The goal of the competition is to find counterfactual explanations for the shortest path problem. This requires deciding what the minimal changes to a road network would make a route chosen by the user the optimal route. This enables explanations such as "Your suggested route would indeed have been optimal, if road X were not a bicycle path." Our solution models the problem as an integer program, iteratively incorporating constraints until an exact solution is found. In the final evaluation on held-out test instances, our method ranked fourth in solution quality and obtained its solution fastest on every instance, with an average runtime of 9.0 seconds compared to 118.8 seconds for the next-fastest submission.

cs.AI↗
arXiv2026

Quantum Query Complexity of Finding a Tarski Fixed Point on a High-Dimensional Grid

The Knaster-Tarski fixed-point theorem states that every monotone function over a complete lattice has a fixed point. Beyond its fundamental role in order theory, the theorem and its algorithmic variants have found broad applications in areas such as economics, game theory, and programming languages. While the query complexity of finding a Tarski fixed point has been extensively studied in classical models, comparatively little is known in the quantum setting. We prove an $Ω(k\log n)$ quantum query lower bound for finding a fixed point of a monotone function on $[n]^k$, using the nonnegative spectral adversary method. In the two extremal regimes $n = 2$ and $k = 1$, our quantum lower bound matches the previous classical lower bounds $Ω(k)$ and $Ω(\log n)$, respectively. For $n, k\geq 2$, our bound improves the best previous classical lower bound when $n < k$ and is within a factor of $\log n / \log k$ compared to the known classical lower bound when $n \geq k$. To construct the adversary matrix, we develop the Tree--Filtration Adversary Method. Besides yielding our lower bound, the method offers a more transparent combinatorial interpretation of the nonnegative spectral adversary method. When the hard instances of a problem admit a tree-like organization and suggest an intuition analogous to classical decision-tree lower bounds, our method provide a promising approach to establishing quantum complexity lower bounds.

quant-ph↗
arXiv2026

Batched Pandora's Box

Motivated by numerous parallelizable stochastic search problems, most notable and timely among them being LLM inference-time scaling, we propose and study batched versions of the Pandora's Box problem of Weitzman. In particular, boxes are opened in capacity-constrained batches, each batch has a setup cost, and all rewards in a batch are revealed together. We consider two different variants, motivated by different application environments: one where boxes are reusable (i.e., can provide multiple i.i.d.~samples) and another where they are not. For both variants we rule out most ``simple'' natural heuristics, and also formally prove NP-hardness of approximation in the traditional sense. We then relax the problem to allow bi-criteria approximations, with respect to both rewards and setup costs, where we exhibit constant approximation algorithms for both the reusable and non-reusable settings. This is obtained through a linear-programming relaxation of Pandora's Box problem, followed by randomized or Pipage rounding.

cs.DS↗
arXiv2026

Diffuse Gaussian Truncation For Deterministic Approximate Counting

We give deterministic FPTASes for two dense counting problems on which the known deterministic algorithms, based on zero-free interpolation, run in quasipolynomial time. For fixed $0<γ<1/2$ and $0<θ\leq1$, the first approximates $\mathrm{haf}(A)$ for a symmetric matrix $A$ when its support graph $G$ has minimum degree at least $(1/2+γ)n$ and its nonzero entries lie in $[θ,1]$. It also approximates permanents under the analogous bipartite condition, including full-support matrices in $[θ,1]$. For fixed $β>0$ and $0<κ\leq1$, the second approximates the zero-field Ising partition function $Z(J)$ for zero-diagonal real symmetric matrices $J$ satisfying $\max_{i,j}|J_{ij}|\leqβ/n$ and $λ_{\max}(J)\leq1-κ$. No separate lower-eigenvalue condition is imposed. We further prove $\log\mathrm{haf}(A)=h_A(G)-n/2+O_{γ,θ}(1)$ and $Z(J)=2^n\det(I-J)^{-1/2}(1+O_{β,κ}(1/n))$. Here $h_A(G)$ is the maximum weighted fractional-matching entropy. For unweighted graphs, the first formula improves the Cuckler--Kahn error from $o(n)$ to $O_γ(1)$ on the fixed-margin class and extends it to weights in $[θ,1]$. Both algorithms use a common Gaussian truncation principle. Each problem becomes an integral of a product of a fixed entire function over Gaussian coordinates, with possibly indefinite moment matrix entries of order $1/n$. Cancelling the linear term and exactly resumming the quadratic term leaves a coordinate remainder vanishing to order at least three. Complex dilation handles small supports. For large supports, we bound the recombined tail by a large-deviation rate that beats the entropy of the subsets. The truncation error is at most $(CR/n)^{R/2}+e^{-cn}$. This faster-than-geometric decay permits $R\log(en/R)=O(\log n+\log(1/ε))$ and hence polynomial enumeration.

cs.DS↗
arXiv2026

Parameterised graph theory for tensor networks: entanglement rerouting, structural simplification, and agnostic tomography

Parameterised graph theory studies how the complexity of graph-theoretic problems depends on structural parameters of the input graph. This perspective has proved useful in analysing tensor-network simulation (Markov and Shi, 2008). Its implications for tensor-network representations and tomography are less well understood. In particular, which graph parameters determine whether a tensor-network state (TNS) admits a tractable matrix product state (MPS) or tree tensor network (TTN) representation, and which control the complexity of learning the state? We address these questions using parameterised graph theory. First, we show that cutwidth and tree-cutwidth bound the bond dimension overhead required to represent a TNS as an MPS or TTN. In the TTN case, tree-cutwidth also bounds the local dimension of the grouped subsystems. The proofs are based on entanglement rerouting, a tensor-network analogue of rerouting information in a classical network. Second, we derive graph-dependent upper bounds on the sample and computational complexity of realisable TNS tomography, with exponents that depend on cutwidth, tree-cutwidth, and a new graph parameter, learning complexity, which we bound in terms of degree and treewidth. We obtain these results by extending the disentangling MPS learner of (Cramer et al., 2010), as analysed further in (Bakshi et al., 2025; Lin et al., 2025), to TTNs and to tensor networks on arbitrary known graphs. Finally, we extend the framework beyond the realisable setting. For an arbitrary input state, our agnostic learner outputs a pure state whose fidelity is within additive error $ε$ of the optimum over tensor-network states on the given graph with a given bond dimension, with explicit graph-dependent bounds on sample and computational complexity.

quant-ph↗
arXiv2026

Computing Lewis Weights to High Precision by Fixed-Point Iteration

The $\ell_p$-Lewis weights of a matrix are defined by a fixed-point equation. For $p<4$, Cohen and Peng [CP15] showed that iterating an equivalent rearrangement of this equation computes Lewis weights to high precision; for $p\geq4$, prior high-precision methods instead use optimization-based approaches. We show that the direct Lewis fixed-point iteration, appearing in the thesis of Lee [Lee16], computes Lewis weights to high precision for every $p>2$. For a matrix $\mathbf{A}\in\mathbb{R}^{m\times n}$ partitioned into row blocks $\mathbf{A}_{[1]},\ldots,\mathbf{A}_{[k]}$, we compute, for $p>2$, coordinatewise $\varepsilon$-approximate $\ell_p$ block Lewis weights in $O\left(p\log\frac{p\sqrt{\sum_{i=1}^k\operatorname{rank}(\mathbf{A}_{[i]})}}{\varepsilon}\right)$ rounds of exact leverage-score-vector computations. For ordinary Lewis weights, this becomes $O\left(p\log\frac{p\sqrt{m}}{\varepsilon}\right)$, improving the $O\left(p^2\log(m/\varepsilon)\right)$ bound of Gribling, Sidford, and Zhang [GSZ26] for $p\geq4$. Our main observation is that each direct Lewis update contracts the KL divergence to the true weights by a factor of $1-\frac{2}{p}$. We also give an alternate explanation of this contraction through volume sampling and entropic independence. Synthetic experiments closely match our predicted local contraction rates and iteration counts grow approximately linearly with $p$; real-data experiments illustrate the information-concentration tradeoff of finite-$p$ block Lewis designs.

cs.DS↗
arXiv2026

Multiobjective Hypergraph Min-Cut in Quasi-Polynomial Time

We study the multiobjective hypergraph min-cut problem: Given a hypergraph $H=(V,E)$ and $k$ cost functions $c_1, c_2, \ldots, c_k:E\to\mathbb{Z}_{\ge 0}$, the goal is to find a non-empty proper subset $U\subsetneq V$ of vertices with minimum $\max_{i\in [k]} c_i(δ(U))$. When $k$ is part of input, the problem is NP-hard (even in graphs). We focus on fixed-constant $k$ setting (e.g., $k=1, 2, 3, \ldots$). Single-objective hypergraph min-cut as well as multiobjective graph min-cut for a constant number of objectives admit polynomial-time algorithms. In contrast to these special cases, the complexity of multiobjective hypergraph min-cut remains open even for $k=2$. Known techniques fail to extend due to structural differences between graphs and hypergraphs. For $k$-objective hypergraph min-cut when $k$ is a fixed constant, we design a randomized PTAS, and two different randomized quasi-polynomial time algorithms. As an application of our $2$-objective hypergraph min-cut results, we obtain a quasi-polynomial time approximation scheme (QPTAS) for hypergraph connectivity interdiction. AI tools were used to iterate and refine the algorithmic ideas underlying this work.

cs.DS↗
arXiv2026

An iterative rounding $2$-approximation for Feedback Vertex Set via AI-assisted proof of an extreme point property

We consider the Feedback Vertex Set problem (FVS): the input is an undirected graph $G=(V,E)$ and the goal is to find a minimum-cardinality (or a min-cost in the weighted case) subset $S \subseteq V$ of vertices such that $G-S$ has no cycles. A $2$-approximation via the local-ratio method was developed in the mid 90's by Bafna, Berman and Fujito (1995) and by Becker and Geiger (1996), and this approximation ratio is tight under UGC. The local-ratio algorithms were later interpreted as primal-dual algorithms via an LP relaxation by Chudak, Goemans, Hochbaum, and Williamson (1998). All known $2$-approximation algorithms for FVS have been via local-ratio and primal-dual methods, and in a quest to obtain a new LP rounding algorithm, it was conjectured (Fiorini 2021) that the Strong-Density polyhedron developed by Chudak, Goemans, Hochbaum, and Williamson has an extreme point property: every basic feasible solution to the LP has a variable with value at least $1/2$. We prove this conjecture. We also consider a related Strong-Edge-Density polyhedron and show the same extreme point property. The advantage of this polyhedron is that it admits a polynomial-time separation oracle and also a compact extended formulation. These results lead to polynomial-time iterative rounding $2$-approximation algorithms. The proof of the extreme point property is of independent technical interest and key ideas in the proof were suggested by AI tools.

cs.DS↗
Compare source metadata on this page
WorkPublishedSource identifierSource
On Maximizing a Weakly Submodular Function over a Matroid Constraint via the Greedy Algorithm2026-09-042609.05008arxiv
Algebraic Geometry Codes Approach the Half-Singleton Bound with Constant Field Size2026-09-042609.05017arxiv
Online Matching in Convex Bipartite Graphs2026-09-042609.05057arxiv
Strategic Facility Location in Euclidean Spaces2026-09-042609.05132arxiv
Machine Unlearning as Private Retroactive Algorithms2026-09-042609.05329arxiv
Integrality Gap Bounds for the Goemans-Linial SDP on Finite Abelian Cayley Graphs2026-09-042609.05368arxiv
Quantum lower bounds for convex optimization and real matrix-vector query problems2026-09-042609.05679arxiv
Approximation algorithms for the square min-sum bin packing problem2026-09-032307.06776arxiv
Weighted Emulators with Local Heaviest Edges Stretch for Undirected Graphs2026-09-032604.26831arxiv
Assortment and Procurement Design in Dual-Mode Content Platforms2026-09-032609.03285arxiv
Counterfactual Routing Using Integer Programming with Constraint Generation2026-09-032609.03707arxiv
Quantum Query Complexity of Finding a Tarski Fixed Point on a High-Dimensional Grid2026-09-032609.03802arxiv
Batched Pandora's Box2026-09-032609.04059arxiv
Diffuse Gaussian Truncation For Deterministic Approximate Counting2026-09-032609.04079arxiv
Parameterised graph theory for tensor networks: entanglement rerouting, structural simplification, and agnostic tomography2026-09-032609.04165arxiv
Computing Lewis Weights to High Precision by Fixed-Point Iteration2026-09-032609.04338arxiv
Multiobjective Hypergraph Min-Cut in Quasi-Polynomial Time2026-09-032609.04389arxiv
An iterative rounding $2$-approximation for Feedback Vertex Set via AI-assisted proof of an extreme point property2026-09-032609.04414arxiv

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