Even Delta-Matroids and the Complexity of Planar Boolean CSPs

TL;DR

This paper generalizes a classical blossom algorithm to efficiently solve certain planar Boolean CSPs involving even Δ-matroid relations, extending tractability to larger classes of Δ-matroids.

Contribution

It introduces an algorithm for edge CSPs with even Δ-matroid constraints and classifies the complexity of planar Boolean CSPs, expanding known tractable classes.

Findings

Efficiently solves Boolean CSPs with even Δ-matroid relations.

Classifies the complexity of planar Boolean CSPs.

Extends tractability to larger classes of Δ-matroids.

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 $\Delta$-matroid relations (represented by lists of tuples). As a consequence of this, we settle the complexity classification of planar Boolean CSPs started by Dvorak and Kupec. Using a reduction to even $\Delta$-matroids, we then extend the tractability result to larger classes of $\Delta$-matroids that we call efficiently coverable. It properly includes classes that were known to be tractable before, namely co-independent, compact, local, linear and binary, with the following caveat: we represent $\Delta$-matroids by lists of tuples, while the last two use a representation by matrices. Since an $n\times n$ matrix can represent exponentially many tuples, our tractability result is not strictly stronger than the known algorithm for linear and binary $\Delta$-matroids.