Theory of periodically specified problems: Complexity and approximability

Abstract

We study the complexity and the efficient approximability of graph and satisfiability problems when specified using various kinds of periodic specifications studied. The general results obtained include the following: (1) We characterize the complexities of several basic generalized CNF satisfiability problems SAT(S) [Sc78], when instances are specified using various kinds of 1- and 2-dimensional periodic specifications. We outline how this characterization can be used to prove a number of new hardness results for the complexity classes DSPACE(n), NSPACE(n), DEXPTIME, NEXPTIME, EXPSPACE etc. These results can be used to prove in a unified way the hardness of a number of combinatorial problems when instances are specified succinctly using various succient specifications considered in the literature. As one corollary, we show that a number of basic NP-hard problems because EXPSPACE-hard when inputs are represented using 1-dimensional infinite periodic wide specifications. This answers a long standing open question posed by Orlin. (2) We outline a simple yet a general technique to devise approximation algorithms with provable worst case performance guarantees for a number of combinatorial problems specified periodically. Our efficient approximation algorithms and schemes are based on extensions of the ideas and represent the first non-trivial characterization of a class of problems having an {epsilon}-approximation (or PTAS) for periodically specified NEXPTIME-hard problems. Two of properties of our results are: (i) For the first time, efficient approximation algorithms and schemes have been developed for natural NEXPTIME-complete problems. (ii) Our results are the first polynomial time approximation algorithms with good performance guarantees for hard problems specified using various kinds of periodic specifications considered in this paper.