Algebraic Global Gadgetry for Surjective Constraint Satisfaction

Abstract

The constraint satisfaction problem (CSP) on a finite relationalstructure B is to decide, given a set of constraints on variableswhere the relations come from B, whether or not there is an assignmentto the variables satisfying all of the constraints; the surjectiveCSP is the variant where one decides the existence of a surjective satisfyingassignment onto the universe of B. We present an algebraicframework for proving hardness results on surjective CSPs; essentially,this framework computes global gadgetry that permits one to presenta reduction from a classical CSP to a surjective CSP. We show how toderive a number of hardness results for surjective CSP in this framework,including the hardness of the disconnected cut problem, of theno-rainbow three-coloring problem, and of the surjective CSP on alltwo-element structures known to be intractable (in this setting). Ourframework thus allows us to unify these hardness results and revealcommon structure among them; we believe that our hardness proof forthe disconnected cut problem is more succinct than the original. Inour view, the framework also makes very transparent a way in whichclassical CSPs can be reduced to surjective CSPs.