Complexity and Efficient Approximability of Two Dimensional Periodically Specified Problems

We 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 us to investigate the complexity and efficient approximability of solving graph theoretic and combinatorial problems when instances are specified using two dimensional periodic specifications. We prove the following results: 1. Several classical NP-hard optimization problems become N EXPTIM E-hard, when in2. In contrast, several of these NEXPTIME-hard problems have polynomial time approxstances are specified using two dimensional periodic speciiications. imakion algorithms with guaranteed worst case performance. Two properties of our results are: 1. For the first time, efficient approximation algorithms and schemes are developed for natural NEXPTIME-complete problems. (Of course it is easy to devise efficient approximation algorithms for arbitrarily “hard” artificial problems.) 2. Om: results are the first polynomial time approximation algorithms with good performance guarantees for “hard” problems specified using any of the kinds of periodic specifications considered here. ‘Current ’4ddress: Los Alamos National Laboratory P.O. Box 1663, MS K990 Los Alamos NM 87545. Email: ‘Department of Computer Science, University at Albany SUNY, Albany, NY 12222. Email addresses of authors: madhavk3 .lnnl.gov. The work is supported by the Department of Energy under Contract W-7405-ENG-36. {hunt, res}@cs .albany. edu. Supported by NSF Grants CCR 90-06396 and CCR94-06611. ,