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版本更新 · v3arXiv:2608.18577

Online Service with Per-Batch Maximum Delays

Tianhang Lu, Runtian Ren, Shengcai Liu, Ke Tang

原文摘要

We study online service with per-batch maximum delays on a finite metric space. Requests arrive over time, and a server serves each batch by following a walk visiting its requests' locations. Movement is instantaneous. The objective is to minimize the total distance traveled by the server plus the sum, over all batches, of the maximum waiting time in each batch. We give deterministic polynomial-time algorithms with competitive ratios $10$ on finite lines, $12$ on weighted trees, and $20$ on arbitrary finite metrics. Unlike additive delay, maximum delay allows younger requests to join a batch without increasing its delay charge. Our algorithms coordinate service using a common waiting clock and a threshold determined by the metric. The analysis uses certificates for groups of requests to compare this policy with any offline schedule, even one that serves different batches and follows different walks. This comparison shows that coordinating service in this way loses only a constant factor. The bounds hold whether the server must serve requests it visits or may leave them unserved; the two rules also have equal offline optimal costs. Offline optimization is polynomial on lines and explicitly represented weighted trees, and NP hard on general metrics. When arrivals are announced but locations are hidden until visited, we give an $84$-competitive algorithm on a known finite line under either rule.

cs.DS

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arXiv ↗PDF ↗
新论文 · v1arXiv:2610.36630

Uniformity Testing for the Monomer-Dimer Model

Joy Qiping Yang, Yuichi Yoshida

原文摘要

We study uniformity testing for the monomer-dimer model on a known graph $G=(V,E)$. Given samples from an unknown distribution $D_w(M)\propto \prod_{e\in M} w_e$ over matchings $M$ of $G$, we ask whether $D_w$ is the uniform distribution over all matchings of $G$ or is $\varepsilon$-far from it in total variation distance. Our main result is that, for every maximum-degree bound $Δ$, uniformity can be tested with $\widetilde O_Δ(\sqrt m/\varepsilon^2)$ samples, where $m=|E|$. We complement this with an information-theoretic lower bound for general graphs: for every pair of parameters $n\le m\le n^2$, there are graphs with $Θ(n)$ vertices and $Θ(m)$ edges on which every tester requires $Ω(\sqrt m/\varepsilon^2)$ samples. Thus, for bounded-degree graph families with $m=Θ(n)$, testing uniformity over the space of matchings has $\widetilde Θ(\sqrt n/\varepsilon^2)$ sample complexity, in contrast with the $Θ(n/\varepsilon^2)$ samples needed to estimate $D_w$ in total variation distance.

cs.DS · cs.DM

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arXiv ↗PDF ↗
新论文 · v1arXiv:2610.36577

An Unconditional Space-Approximation Lower Bound for Semi-Streaming Matching

Yang P. Liu, Kheeran Naidu, Hoai-An Nguyen, Janani Sundaresan

原文摘要

We prove the first unconditional lower bound tradeoff between the number of passes and the approximation factor of maximum matchings in the semi-streaming model. Specifically, we show that any algorithm that outputs an $(1-\varepsilon)$-approximation to maximum matching requires $Ω(\log(1/\varepsilon))$ passes for constant $\varepsilon \in (0,1)$. Prior to our result, such tradeoffs were proven only conditioned on a certain combinatorial assumption about the density of Ruzsa-Szemerédi graphs. The state-of-the-art unconditional lower bound rules out any approximation factor better than $(8/9)$ in two passes but leaves open the possibility of a $(1-\varepsilon)$-approximation for any $\varepsilon > 0$ even in three passes. We provide two different proofs of our result: the first proof uses the recent hierarchical-embedding framework of Assadi, Konrad, Naidu, and Sundaresan (STOC 2024) along with combinatorial constructions using Ruzsa-Szemerédi graphs. The second extends the two-level Ruzsa-Szemerédi hierarchy of Konrad and Naidu (SODA 2024) beyond two levels and proves the corresponding communication lower bound directly by induction, rather than through a reduction to Index.

cs.DS

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arXiv ↗PDF ↗
版本更新 · v3arXiv:2610.00920

A Faster Auction Algorithm for Weighted Matroid Intersection

Tatsuya Terao

原文摘要

We consider the weighted matroid intersection problem in the independence-oracle model. A sequence of works by Huang--Kakimura--Kamiyama~[SODA'16 \& Math. Program'19], Chekuri--Quanrud~[SODA'16], Quanrud~[ICALP'24], and Dudeja--Grilnberger~[IPCO'26] has developed efficient $(1-\varepsilon)$-approximation algorithms for this problem. We present a simple deterministic auction algorithm that, given two matroids on a common ground set of size $n$, computes a $(1-\varepsilon)$-approximate maximum-weight common independent set using $O(n \varepsilon^{-2} \log^2(n))$ independence-oracle queries. This is the first deterministic $(1-\varepsilon)$-approximation algorithm for the weighted matroid intersection problem whose query complexity is nearly linear in $n$ and polynomial in $1/\varepsilon$. Our algorithm builds on the auction algorithm for unweighted matroid intersection by Huang--Kobayashi~['26].

cs.DS

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2609.17533

GPU-Accelerated Search for Fast Matrix Multiplication over $\mathbb{F}_2$

Zeke Medley, Anagha Gokul, Quan Luu, Panagiotis Manolios

原文摘要

We present a GPU-accelerated algorithm for searching for ways to multiply matrices with few scalar multiplications. Our algorithm searches the matrix multiplication flip graph and, on an NVIDIA H200, performs over a billion search steps per second for matrix sizes up to $7\times7$, a roughly $380\times$ improvement over previous GPU-accelerated search on the tensor for $7\times7$ matrix multiplication. We also prove that, over \(\mathbb{F}_2\), the directed matrix multiplication flip graph is strongly connected with only flip and plus edges. This removes the need for computationally expensive reduction edges in the connectivity argument used by previous work. Using this GPU-accelerated search procedure on our simplified flip graph, we find a way to multiply $7\times 7$ matrices over $\mathbb{F}_2$ using 245 multiplications, a three-multiplication improvement over the previous record.

cs.SC · cs.DS

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2602.06860

Single-Range-Cover Search on DAGs: Average-Case Analysis, Efficient Implementations, Privacy Guarantees

Ladan Kian, Dariusz R. Kowalski

原文摘要

Extracting exact and complete information using range queries is not only time-inefficient, on average, but also susceptible to privacy breaches. As a remedy, Demertzis et al.~\cite{demertzis2016practical} (SIGMOD'16) introduced, in the context of searchable symmetric encryption, the principle of covering a query with a single canonical range using a DAG structure (TDAG). It was later formalized by Falzon et al.~\cite{falzon2022range} (VLDB'22) as the single range cover (SRC) search algorithm -- a faster and potentially privacy-preserving search principle. SRC-search can, however, produce false positives. In this work we study how a family of specific Directed Acyclic Graphs (DAGs) could substantially decrease the false-positive ratio, without asymptotic increase of SRC-search time and storage, compared to the classical 1D-Tree. We perform a competitive analysis of a $c$-DAG with respect to the 1D-Tree and derive only an {\bf\em additive constant Returned Level Overhead} while reaching a substantial {\bf\em multiplicative logarithmic improvement in the Accuracy Ratio}, on average. We also provide a generic framework extending our results to non-uniform datasets whose level overhead distribution is close to that of a uniform dataset. We define and analyze several single-path local selection rules that can substantially speed up the original SRC-search algorithm, which was based on exhaustive search, and their privacy implications. Experiments on the Gowalla and NYC Yellow Taxi datasets show that the level overhead stays at or below the proven constant, up to sampling error, and that the level overhead distribution of real data is close to a uniform baseline in most configurations.

cs.DS

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2607.27264

Improved Lower Bounds on Tree Covers

Yu Chen, Shengtang Huang, Zihan Tan, Hangyu Xu

原文摘要

A tree cover of an $n$-point metric space is a collection of $k$ trees such that every pairwise distance is approximately preserved in at least one tree. For fixed $k$, the best known distortion upper bound is $\tilde{O}(n^{1/k})$ and the best known lower bound is $Ω_k(n^{1/2^{k-1}})$. In this paper, we prove a lower bound of $Ω_k(n^{1/(k-1)^2})$ for all $k\ge 3$, significantly narrowing the gap towards the upper bound. Our approach makes use of a novel algebraic topological notion that we call the \emph{agreement index}: the equivariant topological index of a simplicial complex encoding pairwise agreement among tree labels. This notion helps characterize the power and limit of algebraic and topological methods for analyzing tree covers, and we believe it will also find applications to other problems.

math.CO · cs.DS

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arXiv ↗PDF ↗
新论文 · v1arXiv:2610.35859

An $O(n^{0.1537})$-Coloring Algorithm for 3-Colorable Graphs

Kaisei Deguchi, Ken-ichi Kawarabayashi, Hirotaka Yoneda, Masataka Yoneda

原文摘要

We present a polynomial-time randomized algorithm that colors every $n$-vertex $3$-colorable graph with $O(n^{0.1537})$ colors. This improves upon the previous bound of $O(n^{0.1954})$ colors by Bansal, Huang, and Lee (SODA '27). To achieve this result, we introduce a new algorithmic framework for finding a large independent set from a solution to the vector coloring SDP. Combined with the (combinatorial) dense-progress algorithm of Kawarabayashi, Thorup, and Yoneda (STOC '24), this framework yields an $n^{1/6 + o(1)}$-coloring algorithm. Furthermore, we strengthen the SDP-based algorithm specifically for 3-colorable graphs to achieve $O(n^{0.1537})$ colors.

cs.DS

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2608.25197

Exact algorithms for optimal discretization

László Kozma, Junqi Tan

原文摘要

The optimal discretization problem asks, given two disjoint sets of points $R$ and $B$ in the plane, for a minimal family of horizontal and vertical lines that separate the two sets, so that no cell delimited by the lines contains points from both sets. The problem arises as a pre-processing in supervised machine learning, and has received significant attention in parameterized algorithmics. Answering the question raised by Bonnet, Giannopoulos, and Lampis [IPEC 2017] and Froese [PhD thesis, 2018], it was shown by Kratsch, Masařík, Muzi, Pilipczuk, and Sorge [SODA 2021] that optimal discretization admits a fixed-parameter algorithm with running time $2^{\mathcal{O}(k^2 \log k)} \cdot n^{\mathcal{O}(1)}$, where $k$ is the solution size and $n = |R| + |B|$. In this paper we give an algorithm for optimal discretization that runs in time $\mathcal{O}(1.9602^n)$. We also study the related point separation problem that asks to separate all input points by axis-parallel lines. For this problem we obtain an algorithm with runtime $\mathcal{O}(1.8899^n)$. Our guarantees follow from structural observations about bichromatic and monochromatic point sets. To our knowledge, these are the first improvements over the trivial $2^n$ bound for both problems.

cs.DS

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arXiv ↗PDF ↗
版本更新 · v4arXiv:2608.06796

Online Multi-Level Aggregation with Maximum Delay

Tianhang Lu, Runtian Ren, Shengcai Liu, Ke Tang

原文摘要

We study Multi-Level Aggregation with Maximum Delay (MLA-MD) on finite weighted rooted trees. Each service pays for a rooted subtree and for the maximum waiting time among the requests it clears. For the offline problem, we give an exact polynomial-time algorithm that computes both the optimum and an optimal service schedule. For the online problem, we give a deterministic $2$-competitive algorithm and, against an oblivious adversary, a randomized $e/(e-1)$-competitive algorithm. Both competitive ratios are optimal, even on rooted paths. We also extend the offline result and both online guarantees to service systems with submodular costs.

cs.DS · cs.DM · math.OC

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arXiv ↗PDF ↗
新论文 · v1arXiv:2610.35543

Tiny Suffix Arrays

Sharma V. Thankachan

原文摘要

Suffix array and inverse suffix array queries on a text of length $n$ can be answered in $\widetilde{O}(τ^3)$ time using an auxiliary data structure occupying $O(n/τ)$ words, for any $τ\in [1,n]$.

cs.DS · cs.IR

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arXiv ↗PDF ↗
新论文 · v1arXiv:2610.35495

Query Complexity of Learning Axis-Aligned Partitions

Hadley Black, Arya Mazumdar, Barna Saha, Geelon So

原文摘要

This work establishes the query complexity of learning high-dimensional boxes in the exact query learning setting. Let $X \subset \mathbb{R}^d$ be a set of $n$ points, labeled by a collection of $k$ boxes: the label of each point in $X$ is determined by the unique box containing it. A learner aims to exactly recover the labels for every point in $X$ given access to a label oracle. We consider three increasingly-structured settings. In the most general setting, there are no conditions on the boxes, and they may overlap. We show that CAL (Cohn-Atlas-Ladner 1994), a classical active learning algorithm, is minimax-optimal with query complexity $Θ(kd \log (n/kd))$. When the boxes are disjoint, we examine two regimes. In the large-$k$ regime, CAL remains minimax-optimal. For the constant-$k$ regime, an optimized variant of the Halving algorithm (Hegedüs 1995) can beat CAL by a $\log(d)$ factor. Finally, in the most-structured settings, the boxes correspond to the leaves of a constant-sized decision tree. We prove a tight query complexity of $Θ(d + \log n)$ that is additive in $d$ and $\log n$, generalizing a recent result (Black-Larsen-Mazumdar-Saha-So 2026) that establishes the query complexity of learning decision stumps.

cs.DS · cs.CG · cs.LG

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2602.15285

Tensor Decomposition for Non-Clifford Gate Minimization

Kirill Khoruzhii, Patrick Gelß, Sebastian Pokutta

原文摘要

Fault-tolerant quantum computation requires substantial resources for non-Clifford gates. Their count also enters exponentially into the runtime of several classical simulation methods. We present algebraic methods for minimizing the non-Clifford CCZ (Toffoli) count in {CNOT,CCZ} phase circuits using tensor decomposition over $\mathbb{F}_2$. On standard cubic benchmarks, these methods match or improve prior automated Toffoli counts, with most instances optimized in under a minute on a single CPU. With additional computation, the decompositions provide effective initializations for existing T-count methods, improving T counts on several circuits. We also present algebraic methods for CNOT-count optimization at fixed Toffoli count. This gives competitive ancilla-free finite-field multiplication circuits and reduces the CNOT count of XOR AES S-box circuits.

quant-ph · cs.DS

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arXiv ↗PDF ↗
新论文 · v1arXiv:2610.35228

Improved Quasi-Polynomial-Time Approximation for Maximum Independent Set

Zhao Song

原文摘要

We give a deterministic quasi-polynomial-time algorithm for the maximum independent set problem. If an $N$-vertex graph contains an independent set of size $2^{-t}N$, where $t$ is an integer with $1\le t\le\sqrt{\log N}/12$, then our algorithm finds an independent set of size $2^t$ in time $N^{O(t)}$. This gives an approximation ratio of $N/2^{Ω(\sqrt{\log N})}$ in time $N^{O(\sqrt{\log N})}$, improving on the approximation ratio $N/2^{Ω((\log N)^{1/3})}$ obtained by Lovett [Lov26]. A limitation of this form of analysis follows from the integrality gaps of Tulsiani [Tul09]. These gaps imply that a universal guarantee of the form $\operatorname{Las}_{R(t)}(G)\ge2^{-t}N\Longrightarrowα(G)\ge2^t$, at $2^{o(\sqrt{\log N\log\log N})}$ levels, cannot hold over a range extending beyond $t=O(\sqrt{\log N\log\log N})$. Thus, the exponent $1/2$ is the limit of this form of analysis, up to a factor of $\sqrt{\log\log N}$.

cs.DS · cs.CC

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arXiv ↗PDF ↗
新论文 · v1arXiv:2610.35131

From Linear to Polylogarithmic: Dynamic Submodular Maximization over Matroids

Kiarash Banihashem, Samira Goudarzi, MohammadTaghi Hajiaghayi, Peyman Jabbarzade, Morteza Monemizadeh

原文摘要

We study the problem of submodular maximization under a matroid constraint. In this problem, there is a monotone submodular function $f: 2^{N} \rightarrow \mathbb{R}^+$ and a matroid $M(N, I)$, accessible via value and independence oracles and the goal is to find the independent set $S \in I$ maximizing the objective function, i.e., $f(S) = \max_{I \in I} f(I)$. In the fully dynamic setting, the active ground set $N \subseteq N$ evolves through element insertions and deletions. The goal is to maintain a solution $S \subseteq N$ with $S \in I$ and $f(S) = \max_{I \subseteq N,, I \in I} f(I)$ after each update. Two approximation algorithms have been proposed to solve this problem: Dütting et al. developed a $(4+ε)$-approximation algorithm with an expected amortized update time of $O\!\bigl(\frac{k^2}ε \log k \log n \log^3 \frac{k}ε\bigr)$, and Banihashem et al. achieved a similar approximation with a better update time $O\!\bigl(k \log k \log^3 \frac{k}ε\bigr)$. However, these methods both become inefficient when the rank $k$ of the matroid is large (e.g., when $k = n^{o(1)}$). The central open question in this line of work is whether a fully dynamic algorithm with polylogarithmic update time exists. We resolve this question by presenting the first fully dynamic algorithm for submodular maximization over matroids with polylogarithmic update time. Our algorithm maintains a $(4+ε)$-approximate solution with expected amortized update time $poly(\log n, ε^{-1})$, where $n = |N|$. In particular, the update time is independent of the matroid rank $k$, thereby eliminating the dependence on $k$ and settling the open problem.

cs.DS

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2610.32545

Single-Machine Guarantees on Unrelated Machines and Beyond: Proportional Fairness for Nonclairvoyant Polytope Scheduling

Sven Jäger, Alexander Lindermayr, Bart Zondervan

原文摘要

The Polytope Scheduling Problem (PSP) is a general framework for preemptive scheduling in which feasible instantaneous job allocations form a packing polytope. In the nonclairvoyant online setting, Proportional Fairness (PF), a practical allocation rule rooted in market theory, is the leading general-purpose algorithm. On a single machine, PF coincides with weighted Round-Robin, which is $2$-competitive for weighted completion time with uniform release dates (optimal for nonclairvoyant algorithms) and $(2+\varepsilon)$-speed $O(1/\varepsilon)$-competitive for weighted flow time. Known guarantees beyond a single machine are weaker. This raises a central question: Are these heterogeneous environments intrinsically harder than a single machine under nonclairvoyance? We show that they are not: PF retains both single-machine guarantees on the novel class of support-submodular polytopes, which includes unrelated machines and polymatroids. On unrelated machines, this refutes a belief (JACM'18, SODA'25) that PF is not $O(1)$-speed $O(1)$-competitive for weighted flow time. Our main technical contribution is an equivalent characterization: a polytope is support submodular if and only if, for all job weights, the optimal weighted completion time is concave in the processing requirements. This structural insight underlies both guarantees. For general PSP, the picture changes: we give the first lower bound above $2$, showing that every randomized nonclairvoyant algorithm has a competitive ratio of at least $2+10^{-5}$ for weighted completion time, even with uniform release dates. On the positive side, we substantially improve the best known guarantee for weighted completion time from $27$ (SICOMP'26) to $4$ via a new dual-fitting analysis of PF. For weighted flow time, we show that PF achieves the first $O(1)$-speed $O(1)$-competitive nonclairvoyant guarantee for graph scheduling.

cs.DS

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arXiv ↗PDF ↗
新论文 · v1arXiv:2610.35020

Attention under Bounded Key-Query Similarity: Space Complexity and Transformer Anisotropy

Justin Y. Chen, Ying Feng, Piotr Indyk, Michael Kapralov, Ekaterina Kochetkova, Boris Prokhorov

原文摘要

A growing body of work on KV-cache compression aims to reduce the memory requirements of autoregressive generation. These methods compress the stored context while approximately preserving attention computation, enabling generation with reduced memory consumption. Almost tight theoretical results on the worst case space complexity of attention have recently been obtained, but they are achieved by instances where queries can be perfectly aligned with keys. This directly contradicts the observed anisotropy of the key/query geometry in real transformers. In particular, it has been shown empirically that key/query cosine similarities are small even for most correlated pairs, i.e., near-perfect alignment of queries with keys does not happen in practice. In this paper, we study the space complexity of attention approximation under the assumption that key/query cosine similarities belong to a small interval around zero, i.e., a query cannot align too much with any key. We give a nearly space optimal algorithm and a corresponding space lower bound. Surprisingly, the distribution of key/query cosine similarities in our lower bound exhibits the anisotropy property empirically established and studied in the literature. Additionally, we show that the simple uniform sampling algorithm is nearly optimal for a typical range of parameters, supporting its surprising effectiveness on real datasets.

cs.LG · cs.DS

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arXiv ↗PDF ↗
版本更新 · v3arXiv:2108.13061

A New Test for Hamming-Weight Dependencies

David Blackman, Sebastiano Vigna

原文摘要

We describe a new statistical test for pseudorandom number generators (PRNGs). Our test can find bias induced by dependencies among the Hamming weights of the outputs of a PRNG, even for PRNGs that pass state-of-the-art tests of the same kind from the literature, and in particular for generators based on $\mathbf F_2$-linear transformations such as the dSFMT, xoroshiro1024+, and WELL512a.

cs.DS · cs.MS

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arXiv ↗PDF ↗
新论文 · v1arXiv:2610.34892

Order-selection Prophet Inequality (Almost) Goes I.I.D

Bo Peng, Zhihao Gavin Tang

原文摘要

We study the order-selection prophet inequality. A gambler knows the distributions of independent nonnegative random values, chooses the order in which to inspect them, and may irrevocably accept at most one value, at the time it is revealed. Since the order is irrelevant when the values are i.i.d., no algorithm can guarantee more than the optimal i.i.d.\ ratio $Γ_\star\approx0.74544$ of the expected maximum. We give an algorithm with competitive ratio $0.74500001$, less than $4.5\times10^{-4}$ below this bound. The algorithm samples independent arrival times and uses item-dependent thresholds. It is designed so that, conditional on its value, every item is selected with probability equal to a fixed fraction of its probability of being the maximum. We show that the constructions of this form that apply one local rule to every item are exactly those given by an allocation kernel, that is, by a coupling of two uniform random variables. Requiring every rarely winning item to follow the optimal i.i.d.\ threshold path singles out a baseline kernel, for which we prove analytically a ratio $Γ_0\approx0.7389$, defined through a one-dimensional control problem. Our final kernel perturbs this baseline, and its guarantee combines analytic estimates with a rigorous computer-assisted certificate.

cs.DS · cs.GT

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arXiv ↗PDF ↗
新论文 · v1arXiv:2610.34851

When is greedy exact for weighted minimum 3-cut?

Armando Rodado Amaris

原文摘要

The greedy algorithm for minimum 3-cut splits a global minimum-weight cut and then the cheaper of its two sides; it is a tight $4/3$-approximation. We show that it is exact, from every global minimum-weight cut, whenever some optimal 3-partition has two parts joined by total weight at least the weighted edge connectivity $λ$. This extends both theorems of Hochbaum and Shmoys from unweighted planar graphs to all weighted graphs, recovers the $4/3$ bound, and generalizes to minimum $k$-cut with a one-paragraph proof. The remaining hard instances have all pair weights below $λ$; for them we prove a merge lemma for Gomory-Hu sides. Finally, we give a 7-vertex graph with a unique Gomory-Hu tree whose unique optimal 3-partition contains no side of that tree, so the first stage of Kapoor's algorithm does not suffice on its own. The main theorem, the 4/3 bound, the merge lemmas and the counterexample are verified in the Lean 4 proof assistant.

cs.DS

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arXiv ↗PDF ↗
新论文 · v1arXiv:2610.34750

Linear Hashing Is Poissonesque

Vinayak M. Kumar

原文摘要

The Poisson approximation theorem states that when $m$ balls are tossed uniformly and randomly into $n$ bins, the distribution of the bin loads can sometimes be approximated by $n$ independent Poisson random variables of mean $m/n$. We show that when the balls are tossed via a random $\mathbb{F}_2$-linear map, the load distribution is still governed by this Poisson distribution, albeit in a (necessarily) weaker manner. This allows us to show that a random linear map behaves like a fully random map in the following ways: 1. $\textbf{Expected loads.}$ We show that the expected value of the $r$th largest bin is governed by the $r$th largest value among $n$ independent Poisson random variables of mean $m/n$. Specializing to the canonical setting of $m=n$ and $r=1$, our bounds show that the expected maximum load when $n$ balls are hashed into $n$ bins by a random $\mathbb{F}_2$-linear map is no more than the corresponding expectation for a fully random map $\textit{plus}$ $\frac{3}2 + o(1)$. Specializing to $m\gg n\log n$, we show that the expected max load is $(1+o(1))m/n$ in this regime. 2. $\textbf{Concentration.}$ When $m\gg n\log (1/ρ)$ balls are tossed into $n$ bins randomly, we expect with $0.99$ probability that $\textit{every}$ density-$ρ$ subset of bins will contain a $(1\pm o(1) )ρ$-fraction of the balls. We show that this remains the case if the $m$ balls are tossed using a random linear map, and our bound matches the $o(1)$-factor produced by the fully random case. This gives a stronger version of the Leftover Hash Lemma for linear maps. An implication of this is that all bins have load $m/n \pm O(\sqrt{(m/n)\log n})$ with probability $0.99$. We obtain these results by showing that the average of a log-potential on the bin loads is upper bounded by the cumulant generating function of a Poisson random variable of mean $m/n$.

cs.DS · cs.CC · cs.IT · math.PR

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arXiv ↗PDF ↗
新论文 · v1arXiv:2610.34725

The Complexity of Value Optimization in Sporadic Task Systems

Christopher Harbourne, Nathan Fisher, Kecheng Yang, Pontus Ekberg, Sanjoy Baruah, Zhishan Guo

原文摘要

Value-based Task Selection (VTS) asks whether a real-time system can select tasks whose total value reaches a required target while preserving every deadline. We study its threshold decision form for finite collections of independent sporadic tasks scheduled preemptively on one processor. We prove that this problem is $Σ_2^P$-complete for both constrained and arbitrary deadlines, even when all numerical parameters are encoded in unary. The existential choice is the selected task set, while the universal condition requires every processor-demand window to remain feasible. Our hardness reduction already uses constrained-deadline tasks. It represents universal assignments by Chinese Remainder Theorem timestamps and uses polynomial-weight local selector locks to enforce a valid existential assignment. The reduction produces polynomially many tasks, and every generated execution time, deadline, period, task value, and target is polynomially bounded in the source-instance size.

cs.CC · cs.DS

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arXiv ↗PDF ↗
版本更新 · v3arXiv:2610.09972

A deterministic polynomial-time algorithm for the Bilu--Linial signing problem on bipartite graphs

Zhiqiang Xu

原文摘要

We give a deterministic polynomial-time algorithm for the Bilu--Linial signing problem on bipartite graphs. For every finite simple bipartite graph of maximum degree at most an integer $Δ\ge3$, the algorithm assigns signs to its edges so that the signed adjacency matrix has operator norm strictly less than $2\sqrt{Δ-1}$. Our algorithm builds on the randomized recursive repair framework of Jadbabaie, Saberi, and Sra~\cite{JSS26}, with deterministic rules for sign selection and vertex deletion.

cs.DS · cs.CC · math.CO

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arXiv ↗PDF ↗
新论文 · v1arXiv:2610.34656

Coordinating Boundaries: Improved Approximation Algorithms for Unweighted Non-Preemptive Throughput Maximization

Weitian Tong, Yao Xu

原文摘要

We study unweighted nonpreemptive throughput maximization: given jobs with processing times, release times, and deadlines, schedule a maximum-cardinality subset on identical machines. For every fixed $\varepsilon>0$, we give randomized algorithms with expected approximation ratios $17/13+\varepsilon$ in polynomial time and $21/17+\varepsilon$ in pseudopolynomial time. These improve the respective $4/3+\varepsilon$ and $5/4+\varepsilon$ guarantees of Armbruster, Grandoni, Tinguely, and Wiese [STOC'26]. Both results hold for an arbitrary number of identical machines. Our algorithms build on their configuration-LP and matching framework, which partitions time into blocks. For a job whose window meets multiple blocks, its boundary blocks are the first and last such blocks. Crossing boundary pairs, whose endpoints strictly interleave, permit additional assignments to sampled execution slots, while a noncrossing family can be coordinated using consistent joint distributions on at most three blocks. The exchange analysis accounts for fractional job assignments that must be erased before inserting an additional job. For the pseudopolynomial implementation, we project configurations onto selected job identities and use separate configuration distributions with common placement marginals. This retains efficient pricing while enforcing the correlations needed for coordination.

cs.DS

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arXiv ↗PDF ↗
版本更新 · v3arXiv:2602.17811

Faster Parallel Batch-Dynamic Algorithms for Low Out-Degree Orientation

Guy Blelloch, Andrew Brady, Laxman Dhulipala, Jeremy Fineman, Kishen Gowda, Chase Hutton

原文摘要

A low out-degree orientation directs each edge of an undirected graph with the goal of minimizing the maximum out-degree of a vertex. In the parallel batch-dynamic setting, one can insert or delete batches of edges, and the goal is to process the entire batch in parallel with work per edge similar to that of a single sequential update and with span (or depth) for the entire batch that is polylogarithmic. In this paper we present work-efficient parallel batch-dynamic algorithms for maintaining a low out-degree orientation of an undirected graph, both in the amortized and worst-case settings. All results herein achieve polylogarithmic depth; the focus of this paper is on minimizing the work, which varies across results. In the amortized setting, we give a parallel batch-dynamic algorithm that maintains a $O(c)$-orientation in $O(\log n)$ work per update in expectation, where $c$ is a known upper bound on the arboricity over the update sequence. Our algorithm has optimal work in expectation. This result is the parallelization of the classic dynamic orientation algorithm of Brodal and Fagerberg [WADS '99], and, in this setting, is a logarithmic factor faster than the best prior parallel amortized algorithm of Liu et al. [SPAA '22]. In the worst-case setting, we give an $O(c+\log n)$-orientation with worst-case expected work per update $O(\log n)$. This is work-efficient, matching the best known sequential dynamic work of Berglin and Brodal [Algorithmica '20], and implies the existence of an $O(c)$-orientation algorithm with $O(\log^2 n)$ worst-case expected work per update. Both of our algorithms also have deterministic bounds with an additional logarithmic factor in the work. Similar to Berglin and Brodal, by adjusting the parameters of our worst-case algorithm, we achieve tradeoffs where the work is faster than $O(\log n)$ in expectation per update but the max out-degree is worse.

cs.DC · cs.DS

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arXiv ↗PDF ↗
版本更新 · v4arXiv:2503.09762

Optimal regret for dynamic matching with static priorities

Süleyman Kerimov, Pengyu Qian, Mingwei Yang, Sophie H. Yu

原文摘要

We study centralized two-way dynamic matching markets in which agents arrive stochastically over time and wait in queues associated with their types. We focus on static priority policies that assign each agent type a fixed ranking over its neighbors and make online matching decisions using only queue availability. We propose the General Priority ($\mathbf{GP}$) policy, which builds on the Tree Priority ($\mathbf{TP}$) policy introduced in [KAG25]. Under the general position gap condition, $\mathbf{GP}$ selects matches using a nondegenerate optimal basic solution of the fluid relaxation. It prioritizes children over parents within trees and consistently gives clockwise neighbors priority over counterclockwise neighbors on each cycle. We establish the optimal all-time regret scaling $O(ε^{-1})$ for $\mathbf{GP}$ on general two-way matching networks, where the general position gap $ε$ measures the fluid relaxation's distance from degeneracy. In particular, this resolves the conjecture by [KAG25] on the optimal $ε$-dependence for $\mathbf{TP}$.

cs.DS · math.OC · math.PR

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2610.21072

0.4633-Approximation for Online and Offline on-Monotone DR-Submodular Maximization

Vaneet Aggarwal, Donglei Du

原文摘要

We obtain the approximation coefficient $α_0=54440750000/117486474169>0.463378$ for maximizing nonnegative, non-monotone DR-submodular functions over compact convex down-closed subsets of the unit cube, with queries allowed throughout the cube. This improves the previous best coefficient, $4/9$. A coordinatewise map, exponential on small coordinates and linear-fractional on large ones, turns the learner's state, a feasible point, into a feasible action whose value is at least $α_0$ times that of any feasible point, up to a linear term. The proof is analytical and reduces to one-dimensional inequalities. The linear term has an unbiased estimator from one noisy-gradient query, so online maximization reduces to linear learning. Under a uniform bound on the gradients, the expected approximate regret is $O(\sqrt T)$ with projected ascent and $O(T^{3/4})$ with one linear-maximization call per round; with one noisy value per round, it is $O(T^{3/4})$ and $O(T^{7/8})$. Online-to-batch conversion gives offline guarantees. With exact values and gradients, contracting the objective toward a feasible interior point yields an expected $(α_0-\varepsilon)$-approximation for every nonnegative $C^1$ DR-submodular objective, with oracle counts polynomial in $n$ and $\varepsilon^{-1}$ and no gradient or smoothness parameter. Sampling and standard rounding give $(α_0-\varepsilon)$-approximation for submodular set functions under one matroid or a fixed number of knapsack constraints.

cs.DS

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arXiv ↗PDF ↗
版本更新 · v3arXiv:2510.08185

k-SUM Hardness Implies Treewidth-SETH

Michael Lampis

原文摘要

We show that if k-SUM is hard, in the sense that the standard algorithm is essentially optimal, then a variant of the SETH called the Primal Treewidth SETH is true. Formally: if there is an $\varepsilon>0$ and an algorithm which solves SAT in time $(2-\varepsilon)^{tw}|φ|^{O(1)}$, where $tw$ is the width of a given tree decomposition of the primal graph of the input, then there exists a randomized algorithm which solves k-SUM in time $n^{(1-δ)\frac{k}{2}}$ for some $δ>0$ and all sufficiently large $k$. We also establish an analogous result for the k-XOR problem, where integer addition is replaced by component-wise addition modulo $2$. As an application of our reduction we are able to revisit tight lower bounds on the complexity of several fundamental problems parameterized by treewidth (Independent Set, Max Cut, $k$-Coloring). Our results imply that these bounds, which were initially shown under the SETH, also hold if one assumes the k-SUM or k-XOR Hypotheses, arguably increasing our confidence in their validity.

cs.CC · cs.DS

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arXiv ↗PDF ↗
版本更新 · v3arXiv:2405.18754Google Research

GIST: Greedy Independent Set Thresholding for Max-Min Diversification with Submodular Utility

Matthew Fahrbach, Srikumar Ramalingam, Morteza Zadimoghaddam, Sara Ahmadian, Gui Citovsky, Giulia DeSalvo

原文摘要

This work studies a novel subset selection problem called max-min diversification with monotone submodular utility ($\textsf{MDMS}$), which has a wide range of applications in machine learning, e.g., data sampling and feature selection. Given a set of points in a metric space, the goal of $\textsf{MDMS}$ is to maximize $f(S) = g(S) + λ\cdot \texttt{div}(S)$ subject to a cardinality constraint $|S| \le k$, where $g(S)$ is a monotone submodular function and $\texttt{div}(S) = \min_{u,v \in S : u \ne v} \text{dist}(u,v)$ is the max-min diversity objective. We propose the $\texttt{GIST}$ algorithm, which gives a $\frac{1}{2}$-approximation guarantee for $\textsf{MDMS}$ by approximating a series of maximum independent set problems with a bicriteria greedy algorithm. We also prove that it is NP-hard to approximate within a factor of $0.5584$. Finally, we show in our empirical study that $\texttt{GIST}$ outperforms state-of-the-art benchmarks for a single-shot data sampling task on ImageNet.

cs.DS · cs.LG

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