Constraint satisfaction: a personal perspective

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 time algorithms. When the general approach to the classification of constraint satisfaction problems is restricted to graphs and digraphs, one finds that the chordality of graphs plays a crucial role both for the structure of allowed constraints and for the structure of an instance. 1 Computational Complexity and the Diagonalization Barrier The early theory of computation focused on the existence of decision procedures. Gödel’s incompleteness theorem indicates that certain properties of natural numbers cannot be decided. The approach is based on Cantor’s diagonalization method, and produces functions that are not recursively enumerable. The same approach shows that the halting problem for a Turing machine is undecidable. Later studies considered restricting the expressive power of Turing machines by limiting the allowed running time or space. This gives rise to the class P of problems that can be solved by a Turing machine in polynomial time, the class NP of problems that can be solved by a Turing machine in nondeterministic polynomial time, and the class PSPACE of problems that can be solved by a Turing machine in polynomial space. These classes have containments P⊆NP⊆PSPACE, and whether P=NP or even P=PSPACE remains open. Each of these classes contains problems that are computationally hardest, and thus complete for the classes. Under the assumption P6=NP, Ladner [75] showed that there exist problems in NP that are neither in P nor NP-complete, again by means of diagonalization. It can be shown, under the same assumption, that whether a problem in NP is polynomial time solvable or NP-complete is undecidable, from the undecidablity of the halting problem. 1 Electronic Colloquium on Computational Complexity, Report No. 21 (2006)