Approximation Algorithms for PSPACE-Hard Hierarchically and Periodically Specified Problems

TL;DR

This paper demonstrates that many PSPACE-hard problems with hierarchical or periodic specifications admit polynomial time approximation schemes, providing the first such schemes for these classes of problems and answering an open question.

Contribution

It introduces the first polynomial time approximation schemes for PSPACE-hard problems specified hierarchically or periodically, expanding understanding of their approximability.

Findings

Most problems have a (1+ε)-approximation algorithm based on standard algorithms.

Existence of polynomial time approximation schemes for planar instances.

First examples of natural PSPACE-hard problems with PTAS.

Abstract

We study the efficient approximability of basic graph and logic problems in the literature when instances are specified hierarchically as in \cite{Le89} or are specified by 1-dimensional finite narrow periodic specifications as in \cite{Wa93}. We show that, for most of the problems $\Pi$ considered when specified using {\bf k-level-restricted} hierarchical specifications or $k$-narrow periodic specifications the following holds: \item Let $\rho$ be any performance guarantee of a polynomial time approximation algorithm for $\Pi$, when instances are specified using standard specifications. Then $\forall \epsilon > 0$, $ \Pi$ has a polynomial time approximation algorithm with performance guarantee $(1 + \epsilon) \rho$. \item $\Pi$ has a polynomial time approximation scheme when restricted to planar instances. \end{romannum} These are the first polynomial time approximation schemes for PSPACE-hard hierarchically or periodically specified problems. Since several of the problems considered are PSPACE-hard, our results provide the first examples of natural PSPACE-hard optimization problems that have polynomial time approximation schemes. This answers an open question in Condon et. al. \cite{CF+93}.