[论文解读] Fractional homomorphism, Weisfeiler-Leman invariance, and the Sherali-Adams hierarchy for the Constraint Satisfaction Problem

Silvia Butti, Víctor Dalmau|arXiv (Cornell University)|Jul 6, 2021

Constraint Satisfaction and Optimization被引用 7

一句话总结

本文在约束满足问题（CSP）的背景下，建立了分数同态、Weisfeiler-Leman（WL）算法与Sherali-Adams（SA）层级之间的深层联系。证明了SA层级的第一级恰好刻画了分数同构关系，且一个CSP在k维WL下不变当且仅当其可由一个由多态性封闭的分数同态族定义。

ABSTRACT

Given a pair of graphs $\ extbf{A}$ and $\ extbf{B}$, the problems of deciding whether there exists either a homomorphism or an isomorphism from $\ extbf{A}$ to $\ extbf{B}$ have received a lot of attention. While graph homomorphism is known to be NP-complete, the complexity of the graph isomorphism problem is not fully understood. A well-known combinatorial heuristic for graph isomorphism is the Weisfeiler-Leman test together with its higher order variants. On the other hand, both problems can be reformulated as integer programs and various LP methods can be applied to obtain high-quality relaxations that can still be solved efficiently. We study so-called fractional relaxations of these programs in the more general context where $\ extbf{A}$ and $\ extbf{B}$ are not graphs but arbitrary relational structures. We give a combinatorial characterization of the Sherali-Adams hierarchy applied to the homomorphism problem in terms of fractional isomorphism. Collaterally, we also extend a number of known results from graph theory to give a characterization of the notion of fractional isomorphism for relational structures in terms of the Weisfeiler-Leman test, equitable partitions, and counting homomorphisms from trees. As a result, we obtain a description of the families of CSPs that are closed under Weisfeiler-Leman invariance in terms of their polymorphisms as well as decidability by the first level of the Sherali-Adams hierarchy.

研究动机与目标

• 以分数同态为表述，刻画Sherali-Adams层级第一级的表达能力。
• 将已知的1维Weisfeiler-Leman不变性与分数同构之间的等价关系，推广至一般关系结构。
• 通过分数同构与树同态，提供CSP中k维Weisfeiler-Leman不变性的组合刻画。
• 基于多态性与SA层级，对在k-WL下不变的CSP建立完整刻画。
• 统一并推广关于分数同构、均衡划分及从树到任意关系结构的计数同态的已知结果。

提出的方法

• 将关系结构之间的分数同态定义为标准同态的松弛，使用分数赋值。
• 引入k-WL不变CSP的概念，并将其与从树到结构的分数同态的存在性联系起来。
• 从给定的树宽小于k的结构Q构造一个树状结构T，使得对任意结构D，均有Hom(T, D*) = Hom(Q, D)。
• 利用SA层级松弛CSP的整数规划形式化，证明第一级恰好捕捉了分数同构关系。
• 证明一个CSP在k-WL下不变当且仅当其在k元树的分数同态所诱导的多态性下封闭。
• 利用均衡划分与从树出发的计数同态，通过k-WL不变性刻画分数同构。

实验结果

研究问题

• RQ1 在CSP背景下，Sherali-Adams层级的第一级与分数同构之间有何关系？
• RQ2 能否通过分数同构与树同态，组合性地刻画CSP的Weisfeiler-Leman不变性？
• RQ3 k维Weisfeiler-Leman不变性与CSP多态性之间的精确关系是什么？
• RQ4 分数同构在多大程度上将图的古典1-WL等价性推广至任意关系结构？
• RQ5 SA层级能否用于通过逻辑可定义性与多态性封闭性，刻画k-WL的表达能力？

主要发现

• Sherali-Adams层级的第一级恰好刻画了关系结构之间的分数同构关系。
• 一个CSP在k维Weisfeiler-Leman下不变，当且仅当其在k元树的分数同态所诱导的多态性下封闭。
• 两个关系结构A与B之间的分数同构，等价于1-WL算法无法区分它们。
• 从树T到D*（D的k元闭包）存在同态，当且仅当从T到D存在分数同态。
• 结构Q的树宽小于k，当且仅当存在一棵k元f树T，使得对所有结构D，均有Hom(T, D*) = Hom(Q, D)。
• 本文将经典通过1-WL、均衡划分及从树出发的计数同态刻画分数同构的方法，推广至任意关系结构。