Even Delta-Matroids and the Complexity of Planar Boolean CSPs

The main result of this article 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 x n matrix can represent exponentially many tuples, our tractability result is not strictly stronger than the known algorithm for linear and binary Delta-matroids.