Min CSP on Four Elements: Moving Beyond Submodularity

We report new results on the complexity of the valued constraint satisfaction problem (VCSP). Under the unique games conjecture, the approximability of finite-valued VCSP is fairly well-understood. However, there is yet no characterisation of VCSPs that can be solved exactly in polynomial time. This is unsatisfactory, since such results are interesting from a combinatorial optimisation perspective; there are deep connections with, for instance, submodular and bisubmodular minimisation. We consider the Min and Max CSP problems (i.e. where the cost functions only attain values in {0,1}) over four-element domains and identify all tractable fragments. Similar classifications were previously known for two- and three-element domains. In the process, we introduce a new class of tractable VCSPs based on a generalisation of submodularity. We also extend and modify a graph-based technique by Kolmogorov and Zivny (originally introduced by Takhanov) for efficiently obtaining hardness results in our setting. This allow us to prove the result without relying on computer-assisted case analyses (which otherwise are fairly common when studying the complexity and approximability of VCSPs.) The hardness results are further simplified by the introduction of powerful reduction techniques.