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Treewidth-Pliability and PTAS for Max-CSPs
arXiv - CS - Discrete Mathematics Pub Date : 2019-11-08 , DOI: arxiv-1911.03204
Miguel Romero, Marcin Wrochna, Stanislav \v{Z}ivn\'y

We identify a sufficient condition, treewidth-pliability, that gives a polynomial-time approximation scheme (PTAS) for a large class of Max-2-CSPs parametrised by the class of allowed constraint graphs (with arbitrary constraints on an unbounded alphabet). Our result applies more generally to the maximum homomorphism problem between two rational-valued structures. The condition unifies the two main approaches for designing PTASes. One is Baker's layering technique, which applies to sparse graphs such as planar or excluded-minor graphs. The other is based on Szemer\'{e}di's regularity lemma and applies to dense graphs. We extend the applicability of both techniques to new classes of Max-CSPs. Treewidth-pliability turns out to be a robust notion that can be defined in several equivalent ways, including characterisations via size, treedepth, or the Hadwiger number. We show connections to the notions of fractional-treewidth-fragility from structural graph theory, hyperfiniteness from the area of property testing, and regularity partitions from the theory of dense graph limits. These may be of independent interest. In particular we show that a monotone class of graphs is hyperfinite if and only if it is fractionally-treewidth-fragile and has bounded degree.

中文翻译:

Max-CSP 的树宽柔度和 PTAS

我们确定了一个充分条件,即树宽柔度,它为一大类 Max-2-CSP 提供了多项式时间近似方案 (PTAS),这些 Max-2-CSP 由允许的约束图类(对无界字母表具有任意约束)参数化。我们的结果更普遍地适用于两个有理值结构之间的最大同态问题。该条件统一了设计 PTAS 的两种主要方法。一种是 Baker 的分层技术,它适用于稀疏图,例如平面图或排除次要图。另一种是基于 Szemer\'{e}di 的正则性引理,适用于稠密图。我们将这两种技术的适用性扩展到新类别的 Max-CSP。Treewidth-pliability 被证明是一个强大的概念,可以用几种等效的方式定义,包括通过大小、treedepth 或 Hadwiger 数。我们展示了与结构图理论中的分数树宽脆弱性概念、属性测试领域中的超有限性以及稠密图限制理论中的规律性划分的概念的联系。这些可能具有独立的利益。特别地,我们证明了单调类图是超有限的,当且仅当它是分数树宽脆弱的并且具有有界度。
更新日期:2020-10-13
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