Weisfeiler-Leman Invariant Promise Valued CSPs

TL;DR

This paper extends the connection between linear programming relaxations and Weisfeiler-Leman invariance from classical CSPs to the broader class of Promise Valued CSPs, revealing new insights into their solvability and the limitations of existing relaxations.

Contribution

It generalizes previous results to Promise Valued CSPs and demonstrates that two standard LP relaxations differ in this broader context.

Findings

LP relaxation equivalence fails for Promise Valued CSPs

Set of Yes instances is closed under Weisfeiler-Leman equivalence

Characterization of solvability via distributed networks

Abstract

In a recent line of work, Butti and Dalmau have shown that a fixed-template Constraint Satisfaction Problem is solvable by a certain natural linear programming relaxation (equivalent to the basic linear programming relaxation) if and only if it is solvable on a certain distributed network, and this happens if and only if its set of Yes instances is closed under Weisfeiler-Leman equivalence. We generalize this result to the much broader framework of fixed-template Promise Valued Constraint Satisfaction Problems. Moreover, we show that two commonly used linear programming relaxations are no longer equivalent in this broader framework.