一句话：将稀疏图上有限值 Max-CSP 的 PTAS 推广到含严格约束的一般值 CSP，并刻画对角化条件。
本文研究稀疏图类上约束满足问题（CSP）的多项式时间近似方案（PTAS），如最大独立集和最小顶点覆盖。此前，Sherali-Adams LP 松弛已对 fractionally-treewidth-fragile 类上的最大化有限值 CSP 给出简单 PTAS。作者将结果推广到一般值 CSP，允许必须满足的严格（crisp）约束，且仅要求其域中存在一个可行性不低于其他元素、但价值可能更低的元素。对于带严格约束的最小化一般值 CSP，论文在 Dvořák 定义的 Baker 图类上构造 PTAS，该图类涵盖已知 Baker 技术适用的类（fractionally-treewidth-fragile 类除外）。通常需要严格约束满足单调性，作者将其放宽为对角化（diagonalisability），这是一种与逻辑、统计物理和随机 CSP 相关的关系结构性质。
We study polynomial-time approximation schemes (PTASes) for constraint satisfaction problems (CSPs) such as Maximum Independent Set or Minimum Vertex Cover on sparse graph classes.Baker's approach gives a PTAS on planar graphs, excluded-minor classes, and beyond. For Max-CSPs, and even more generally, maximisation finite-valued CSPs (where constraints are arbitrary non-negative functions), Romero, Wrochna, and Živný [SODA'21] showed that the Sherali-Adams LP relaxation gives a simple PTAS for all fractionally-treewidth-fragile classes, which is the most general "sparsity" condition for which a PTAS is known. We extend these results to general-valued CSPs, which include "crisp" (or "strict") constraints that have to be satisfied by every feasible assignment. The only condition on the crisp constraints is that their domain contains an element which is at least as feasible as all the others (but possibly less valuable).For minimisation general-valued CSPs with crisp constraints, we present a PTAS for all Baker graph classes - a definition by Dvořák [SODA'20] which encompasses all classes where Baker's technique is known to work, except for fractionally-treewidth-fragile classes. While this is standard for problems satisfying a certain monotonicity condition on crisp constraints, we show this can be relaxed to diagonalisability - a property of relational structures connected to logics, statistical physics, and random CSPs.
DOI 原文 ·
@inproceedings{paperbot6226,
title = {PTAS for Sparse General-Valued CSPs},
author = {Balazs F. Mezei and Marcin Wrochna and Stanislav Zivny},
booktitle = {LICS},
year = {2021},
doi = {10.1109/lics52264.2021.9470599}
}