Classification of Bipartite Boolean Constraint Satisfaction through Delta-Matroid Intersection

Even Delta-Matroids and the Complexity of Planar Boolean CSPs

“…The following theorem is a strengthening of a result from [12]: Theorem 2.3. Given an edge CSP instance I with effectively coverable ∆-matroid constraints, an optimal edge labeling f of I can be found in time polynomial in |I|.…”

“…The whole construction is similar to, but more general than, C-zebra ∆-matroids introduced in [12]. We note here also that the class of coverable ∆-matroids is natural in the sense of being closed under direct products and identifying variables (in other words, gadget constructions).…”

“…As we show in Appendix D, effectively coverable ∆-matroid classes include numerous known tractable classes of ∆-matroids: C-zebra ∆-matroids [12] for any C subclass of even ∆-matroids (where we assume, just like in [12], that we are given the zebra representations on input) as well as co-independent [10], compact [15], local [7], and binary [13,7] ∆-matroids. To our best knowledge these are all the known tractable classes and according to [7] the classes other than C-zebras are pairwise incomparable.…”

“…The idea of the algorithm is similar to what [12] previously did for C-zebra ∆-matroids, but our method covers a larger class of ∆-matroids.…”

“…Since then, there has also been progress on the algorithmic side. Several tractable (in the sense of being polynomial time solvable) classes of ∆-matroids were identified [10,12,15,7,13,8]. A recurring theme is the connection between ∆-matroids and matching problems.…”