Optimal Inapproximability with Universal Factor Graphs

IF 0.9 3区计算机科学 Q3 COMPUTER SCIENCE, THEORY & METHODS ACM Transactions on Algorithms Pub Date : 2023-12-15 DOI:10.1145/3631119

Per Austrin, Jonah Brown-Cohen, Johan Håstad

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Abstract

The factor graph of an instance of a constraint satisfaction problem (CSP) is the bipartite graph indicating which variables appear in each constraint. An instance of the CSP is given by the factor graph together with a list of which predicate is applied for each constraint. We establish that many Max-CSPs remain as hard to approximate as in the general case even when the factor graph is fixed (depending only on the size of the instance) and known in advance.

Examples of results obtained for this restricted setting are:

(1)	Optimal inapproximability for Max-3-Lin and Max-3-Sat (Håstad, J. ACM 2001).
(2)	Approximation resistance for predicates supporting pairwise independent subgroups (Chan, J. ACM 2016).
(3)	Hardness of the “(2 + ϵ)-Sat” problem and other Promise CSPs (Austrin et al., SIAM J. Comput. 2017).

The main technical tool used to establish these results is a new way of folding the long code which we call “functional folding”.

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通用因子图的最优不可逼近性

约束条件满足问题（CSP）实例的因子图是表示每个约束条件中出现的变量的双向图。CSP 实例由因子图和每个约束条件所应用的谓词列表给出。我们发现，即使因子图是固定的（只取决于实例的大小）并且事先已知，许多 Max-CSP 仍然和一般情况下一样难以近似。在这种受限情况下获得的结果举例如下(1)Max-3-Lin 和 Max-3-Sat 的最优不可逼近性（Håstad，J. ACM 2001）。(2)支持成对独立子群的谓词的逼近阻力（Chan，J. ACM 2016）。(3)"(2 + ϵ)-Sat "问题和其他 Promise CSP 的硬度（Austrin et al、用于建立这些结果的主要技术工具是一种折叠长代码的新方法，我们称之为 "函数折叠"。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

ACM Transactions on Algorithms COMPUTER SCIENCE, THEORY & METHODS-MATHEMATICS, APPLIED

CiteScore

3.30

自引率

0.00%

发文量

审稿时长

6-12 weeks

期刊介绍： ACM Transactions on Algorithms welcomes submissions of original research of the highest quality dealing with algorithms that are inherently discrete and finite, and having mathematical content in a natural way, either in the objective or in the analysis. Most welcome are new algorithms and data structures, new and improved analyses, and complexity results. Specific areas of computation covered by the journal include combinatorial searches and objects; counting; discrete optimization and approximation; randomization and quantum computation; parallel and distributed computation; algorithms for graphs, geometry, arithmetic, number theory, strings; on-line analysis; cryptography; coding; data compression; learning algorithms; methods of algorithmic analysis; discrete algorithms for application areas such as biology, economics, game theory, communication, computer systems and architecture, hardware design, scientific computing

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