A New Hybrid Tractable Class of Soft Constraint Problems

The constraint satisfaction problem (CSP) is a central generic problem in artificial intelligence. Considerable effort has been made in identifying properties which ensure tractability in such problems. In this paper we study hybrid tractability of soft constraint problems; that is, properties which guarantee tractability of the given soft constraint problem, but properties which do not depend only on the underlying structure of the instance (such as being tree-structured) or only on the types of soft constraints in the instance (such as submodularity). We firstly present two hybrid classes of soft constraint problems defined by forbidden subgraphs in the structure of the instance. These classes allow certain combinations of binary crisp constraints together with arbitrary unary soft constraints. We then introduce the joint-winner property, which allows us to define a novel hybrid tractable class of soft constraint problems with soft binary and unary constraints. This class generalises the SoftAllDiff constraint with arbitrary unary soft constraints. We show that the jointwinner property is easily recognisable in polynomial time and present a polynomial-time algorithm based on maximum-flows for the class of soft constraint problems satisfying the joint-winner property. Moreover, we show that if cost functions can only take on two distinct values then this class is maximal.