Complexity of hierarchically and 1-dimensional periodically specified problems I: Hardness results

Abstract

We study the complexity of various combinatorial and satisfiability problems when instances are specified using one of the following specifications: (1) the 1-dimensional finite periodic narrow specifications of Wanke and Ford et al. (2) the 1-dimensional finite periodic narrow specifications with explicit boundary conditions of Gale (3) the 2-way infinite1-dimensional narrow periodic specifications of Orlin et al. and (4) the hierarchical specifications of Lengauer et al. we obtain three general types of results. First, we prove that there is a polynomial time algorithm that given a 1-FPN- or 1-FPN(BC)specification of a graph (or a C N F formula) constructs a level-restricted L-specification of an isomorphic graph (or formula). This theorem along with the hardness results proved here provides alternative and unified proofs of many hardness results proved in the past either by Lengauer and Wagner or by Orlin. Second, we study the complexity of generalized CNF satisfiability problems of Schaefer. Assuming P {ne} PSPACE, we characterize completely the polynomial time solvability of these problems, when instances are specified as in (1), (2),(3) or (4). As applications of our first two types of results, we obtain a number of new PSPACE-hardness and polynomial time algorithms for problems specified as in (1), (2), (3) or(4). Many of our results also hold for O(log N) bandwidth bounded planar instances.