The complexity of approximating PSPACE-Complete problems for hierarchical specifications

We extend the concept of polynomial time approximation algorithms to apply to problems for hierarchically speciied graphs, many of which are PSPACE-complete. Assuming P 6 = PSPACE, the existence or nonexistence of such eecient approximation algorithms is characterized, for several standard graph theoretic and combinatorial problems. We present polynomial time approximation algorithms for several standard PSPACE-hard problems considered in the literature. In contrast, we show that unless P = PSPACE, there is no polynomial time-approximation for any > 0, for several other problems, when the instances are speciied hierarchically. We present polynomial time approximation algorithms for the following problems when the graphs are speciied hierarchically: minimum vertex cover, maximum 3SAT, weighted max cut, minimum maximal matching, and bounded degree maximum independent set. In contrast, we show that unless P = PSPACE, there is no polynomial time-approximation for any > 0, for the following problems when the instances are speciied hierarchically: the number of true gates in a monotone acyclic circuit when all input values are speciied and the optimal value of the objective function of a linear program. It is also shown that unless P = PSPACE, a performance guarantee of less than 2 cannot be obtained in polynomial time for the following problems when the instances are speciied hierarchically: high degree subgraph, k-vertex connected subgraph and k-edge connected subgraph.