Complexity and efficient approximability of two dimensional periodically specified problems

The authors consider the two dimensional periodic specifications: a method to specify succinctly objects with highly regular repetitive structure. These specifications arise naturally when processing engineering designs including VLSI designs. These specifications can specify objects whose sizes are exponentially larger than the sizes of the specification themselves. Consequently solving a periodically specified problem by explicitly expanding the instance is prohibitively expensive in terms of computational resources. This leads one to investigate the complexity and efficient approximability of solving graph theoretic and combinatorial problems when instances are specified using two dimensional periodic specifications. They prove the following results: (1) several classical NP-hard optimization problems become NEXPTIME-hard, when instances are specified using two dimensional periodic specifications; (2) in contrast, several of these NEXPTIME-hard problems have polynomial time approximation algorithms with guaranteed worst case performance

Approximation algorithms for NEXTtime-hard periodically specified problems and domino problems

We study the efficient approximability of two general class of problems: (1) optimization versions of the domino problems studies in [Ha85, Ha86, vEB83, SB84] and (2) graph and satisfiability problems when specified using various kinds of periodic specifications. Both easiness and hardness results are obtained. Our efficient approximation algorithms and schemes are based on extensions of the ideas. Two of properties of our results obtained here are: (1) For the first time, efficient approximation algorithms and schemes have been developed for natural NEXPTIME-complete problems. (2) 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. Our results significantly extend the results in [HW94, Wa93, MH+94]

Complexity of hierarchically and 1-dimensional periodically specified problems

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

On Approximation Algorithms for Hierarchical MAX-SAT

We prove upper and lower bounds on performance guarantees of approximation algorithms for the Hierarchical MAX-SAT (H-MAX-SAT) problem. This problem is representative of a broad class of PSPACE-hard problems involving graphs, Boolean formulas and other structures that are defined succinctly. Our first result is that for some constant ffl ! 1, it is PSPACE-hard to approximate the function H-MAX-SAT to within ratio ffl. We obtain our result using a reduction from the language recognition problem for a model of PSPACE called the probabilistically checkable debate system. As an immediate application, we obtain nonapproximability results for functions on hierarchical graphs by combining our result with previously known approximation-preserving reductions to other problems. For example, it is PSPACE-hard to approximate H-MAX-CUT and H-MAX-INDEPENDENT-SET to within some constant factor. Our second result is that there is an efficient approximation algorithm for H-MAX-SAT with performance guar..