Constraint satisfaction: a personal perspective.
Tomás Feder · 2006
Attempts at classifying computational problems as polynomial time solvable, NP-complete, or belonging to a higher level in the polynomial hierarchy, face the difficulty of undecidability. These classes, including NP, admit a logic formulation. By suitably restricting the formulation, one finds the logic class MMSNP, or monotone monadic strict NP without inequality, as a largest class that seems to avoid diagonalization arguments. Representative of this logic class is the class CSP of constraint satisfaction problems. Both MMSNP and CSP admit generalizations via alternations of quantifiers corresponding to higher levels in the hierarchy. Examining CSP from a computational point of view, one finds that the polynomial time solvable problems that do not have the bounded width property of Datalog are group theoretic in nature. In general, closure properties of the constraints characterize the complexity of the problems. When one restricts the number of occurrences of each variable, the problems that are encountered relate to deltamatroid intersection. When such a restriction forbidding copying is introduced in the context of input-output constraints, one finds nonexpansive mappings as characterizing this restriction. Both delta-matroid intersection and nonexpansive network stability problems yield polynomial