The Complexity of Approximation PSPACE-Complete Problems for Hierarchical Specifications

We extend the concept of polynomial time approximation algorithms to apply to problems for hierarchically specified graphs, many of which are PSPACE-complete. We present polynomial time approximation algorithms for several standard PSPACE-hard problems considered in the literature. In contrast, we prove that finding e-approximations for any e > 0, for several other problems when the instances are specified hierarchically, is PSPACE-hard. We present polynomial time approximation algorithms for the following problems when the graphs are specified hierarchically: minimum vertex cover, maximum 3SAT, weighted max cut, minimum maximal matching, and bounded degree maximum independent set.In contrast, we show that for any e > 0, obtaining e-approximations for the following problems when the instances are specified hierarchically is PSPACE-hard: the number of true gates in a monotone acyclic circuit when all input values are specified and the optimal value of the objective function of a linear program. It is also shown that obtaining a performance guarantee of less than 2 is PSPACE-hard for the following problems when the instances are specified hierarchically: high degree subgraph, k-vertex connected subgraph and k-edge connected subgraph.

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