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arXiv · 2401.16998

The Sherali-Adams and Weisfeiler-Leman hierarchies in (Promise Valued) Constraint Satisfaction Problems

Libor Barto·Silvia Butti·Víctor Dalmau
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Abstract

In this paper we study the interactions between so-called fractional relaxations of the integer programs (IPs) which encode homomorphism and isomorphism of relational structures. We give a combinatorial characterization of a certain natural linear programming (LP) relaxation of homomorphism in terms of fractional isomorphism. As a result, we show that the families of constraint satisfaction problems (CSPs) that are solvable by such linear program are precisely those that are closed under an equivalence relation which we call Weisfeiler-Leman invariance. We also generalize this result to the much broader framework of Promise Valued Constraint Satisfaction Problems, which brings together two well-studied extensions of the CSP framework. Finally, we consider the hierarchies of increasingly tighter relaxations of the homomorphism and isomorphism IPs obtained by applying the Sherali-Adams and Weisfeiler-Leman methods respectively. We extend our combinatorial characterization of the basic LP to higher levels of the Sherali-Adams hierarchy, and we generalize a well-known logical characterization of the Weisfeiler-Leman test from graphs to relational structures.

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Libor Barto, Silvia Butti, Víctor Dalmau. 2024-01-30. The Sherali-Adams and Weisfeiler-Leman hierarchies in (Promise Valued) Constraint Satisfaction Problems. https://arxiv.org/abs/2401.16998

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