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版本更新 · v6arXiv:2508.05597

确定性通信复杂度的NP困难性与加性不可近似性

NP-Hardness and Additive Inapproximability of Deterministic Communication Complexity

Serge Gaspers, Tao Zixu He, Simon Mackenzie

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作者报告：给定布尔函数完整真值表，确定两方计算该函数所需最小最坏情况通信比特数是NP困难的，且在任何固定加性误差下不可近似。

cs.CC · cs.DS · math.CO

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Given the full truth table of a Boolean function whose input is split between two players, we study the complexity of determining the minimum worst-case number of bits they must communicate to compute it. This is a basic meta-complexity question, posed by Yao in 1979. We prove NP-hardness of approximation within every fixed additive error for deterministic communication complexity, under deterministic polynomial-time reductions. The result holds both in the standard model and in Yao's original model, where the players alternate after every bit and only the receiver of the last bit must determine the answer. In particular, it resolves the exact computation question under Yao's original conventions. Earlier versions of this same paper established exact NP-hardness for the standard model, independently and concurrently with Hirahara, Ilango, and Loff. In both models, a polynomial-time estimator with any fixed additive error would yield a polynomial-time $3$-SAT algorithm, deterministic or bounded-error randomized according to the estimator. For the standard model, we also give quantitative implications for errors that grow with the number $T$ of table entries. A bounded-error randomized polynomial-time estimator with additive error $c\log T/\log\log T$, for any fixed $0\le c<1/8$, yields a bounded-error randomized $3$-SAT algorithm on $N$ variables and $O(N)$ clauses running in time $2^{O(N^α)}$ for every fixed $8c<α<1$. A deterministic polynomial-time estimator with additive error $c_*\log\log T$, for an effective absolute constant $c_*>0$, yields a deterministic algorithm running in time $2^{O((\log N)^2)}$ on such formulas.

新论文 · v1arXiv:2611.05146

Path Decompositions of Sparse Graphs via Gaussian Sweeps and Faster Exponential Algorithms

Serge Gaspers

原文摘要

Using the bisection bound of Monien and Preis, Fomin and Høie proved in 2006 that every graph of maximum degree 3 on $n$ vertices has pathwidth at most $n/6+o(n)$. We prove that the pathwidth, bisection width and cutwidth of these graphs are at most $c_3 n+o(n)$, where $c_3=\frac{3}{4π}\arccos\frac56\approx0.1398$. For bisection width this improves the bound $n/6+o(n)$ of Monien and Preis (2006) and extends to all such graphs the bound that Lichev and Mitsche (2023) obtained for random cubic graphs. For maximum degree 4, we prove that bisection width and cutwidth are at most $n/3+o(n)$, improving the bound $2n/5+o(n)$ of Monien and Preis for 4-regular graphs, and that pathwidth is at most $0.2458n$, improving the bound $n/3$ of Fomin, Gaspers, Saurabh and Stepanov (2009). We show that every graph with $m$ edges has pathwidth at most $7m/48+o(n)$, improving the bound $13m/75$ of Scott and Sorkin (2007), and we give further bounds of the form $\sum_{d\ge3}a_d n_d+o(n)$ with $a_d<1$, where $n_d$ is the number of vertices of degree $d$. The path decompositions, and linear orderings of the stated cutwidth, can be computed in polynomial time. Instead of deriving pathwidth bounds from bisection width, we give each vertex a Gaussian label, a weighted sum of independent standard normal variables over a ball around it, and sweep a threshold through the labels; the bags are frontiers, the vertices above the threshold with a neighbor below it. We bound the probability that a vertex lies in a frontier by comparison with an explicit Gaussian model of a vertex and its neighbors. Faster exact algorithms follow: for feedback vertex set and the traveling salesperson problem on graphs of bounded maximum or average degree, and on general graphs for minimum weight dominating set (by branching until the instance is sparse), Max 2-CSP and counting subgraphs with $k$ edges (by the bound in $m$).

cs.DS · cs.DM · math.CO

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新论文 · v1arXiv:2611.01774

Monotone Local Search with Sample-Dependent Running Times

Serge Gaspers

原文摘要

Monotone Local Search turns an algorithm that extends a given partial solution into an algorithm that finds a solution from scratch: sample a few elements, hope that they lie in an unknown solution, and complete them. Its known analyses bound every call of the extension algorithm by a worst-case time that depends only on the number of sampled elements, and Esmer et al. (SODA 2024) showed that no algorithm that uses the extension algorithm as a black box with this running time beats the original form by more than a polynomial factor. We let this running time depend on the actual sample. If the extension algorithm decides in time $O^*(2^{g(I,X,k)})$, for a polynomial-time computable $g$, whether a sample $X$ can be completed to a solution with at most $k$ elements, our main theorem gives a randomized algorithm with one-sided error in time $O^*(2^{ψn})$ whenever, for some solution $S$ of size $k$ and some $t$, the exponential moment $\sum_{X\subseteq S,|X|=t}2^{-g(I,X,k)}$ is at least $2^{-ψn}\binom{n}{t}$. A corollary, multivariate Monotone Local Search, generalizes the theorems of Fomin et al. (STOC 2016, J. ACM 2019) and of Gaspers and Lee (ICALP 2017). For Max Leaf Spanning Tree and, equivalently, Connected Dominating Set, the previous best polynomial-space bound is $O^*(1.8619^n)$. Completing a sample is a Steiner tree problem, solved in time exponential in the number of terminals, which varies strongly with the sample. We obtain a randomized polynomial-space algorithm in time $O(1.6991^n)$ and a deterministic one in time $(\sqrt5-\frac12)^n 2^{o(n)}=O(1.7361^n)$. For connectivity problems in which the extension algorithm's running time doubles with every component of the sample, an analysis through a partition function gives randomized polynomial-space algorithms in time $O^*(1.8^n)$ for Connected Feedback Vertex Set and $O(1.6444^n)$ for Connected Vertex Cover.

cs.DS · cs.DM

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