Even Delta-Matroids and the Complexity of Planar Boolean CSPs

Abstract

The main result of this paper is a generalization of the classical blossom algorithm for finding perfect matchings. Our algorithm can efficiently solve Boolean CSPs where each variable appears in exactly two constraints (we call it edge CSP) and all constraints are even ∆-matroid relations (represented by lists of tuples). As a consequence of this, we settle the complexity classification of planar Boolean CSPs started by Dvořák and Kupec.Knowing that edge CSP is tractable for even ∆-matroid constraints allows us to extend the tractability result to a larger class of ∆-matroids that includes many classes that were known to be tractable before, namely co-independent, compact, local and binary.

On planar valued CSPs

“…For valued constraint language on Boolean domains, we have given a necessary condition for tractability. The obvious open problem is to give a complexity classification of Boolean valued constraint languages, following a classification of crisp Boolean constraint languages [12,26]. Another line of work is to consider larger domains in the non-conservative setting.…”

“…Planar restrictions have been studied for Boolean (decision) CSPs [12,26], for Boolean symmetric counting CSPs with real [5] and complex [20] weights, and also for Boolean CSPs with respect to polynomial-time approximation schemes [27,10].…”