The Complexity of Boolean Surjective General-Valued CSPs

Valued constraint satisfaction problems (VCSPs) are discrete optimisation problems with a Q-valued objective function given as a sum of fixed-arity functions, where Q = Q ∪ {∞} is the set of extended rationals. In Boolean surjective VCSPs variables take on labels from D = {0, 1} and an optimal assignment is required to use both labels from D. A classic example is the global mincut problem in graphs. Building on the work of Uppman, we establish a dichotomy theorem and thus give a complete complexity classification of Boolean surjective VCSPs. The newly discovered tractable case has an interesting structure related to projections of downsets and upsets. Our work generalises the dichotomy for {0,∞}-valued constraint languages (corresponding to CSPs) obtained by Creignou and Hébrard, and the dichotomy for {0, 1}-valued constraint languages (corresponding to Min-CSPs) obtained by Uppman.