# On the Efficient Approximability of “HARD” Problems: A Survey

## Abstract

By a **HARD** problem, we mean a problem that is **PSPACE**-, **DEXPTIME**-, **NEXPTIME**-hard, etc. Many basic algorithmically-solvable problems, for quantified formulas, for sequential circuits, for combinatorial games, and for problems when instances are specified hierarchically or periodically are known to be **HARD**. Analogous to what has occurred for **NP**-complete problems, it often makes sense to talk about the complexities of the approxi-mation problems associated with these **HARD** problems. Here, we survey our results on the complexities of such approximation problems, emphasizing our results for hierarchically-and periodically-specified problems. These results include the first collection of PTASs, for *natural* **PSPACE**-complete, **DEXPTIME**-complete, and **NEXPTIME**-complete problems in the literature. In contrast, these results also include a number of new results showing that related approximation problems are **HARD**.

- 1.
“Local” approximation-preserving reductions between problems can be extended to efficient approximation-preserving reductions between these problems, when instances are hierarchically- or periodically-specified. Such reductions can be used both to obtain efficient approximation algorithms and to prove approximation problems are

**HARD**. - 2.
Hierarchically- and periodically-specified problems are often

**HARD**. But, they also are often efficiently approximable. - 3.
The efficient decomposability of problems and problem instances and the compatibility of such decompositions with the structure of hierarchical- or periodic-specifications play central roles in the development of efficient approximation algorithms, for the problems when hierarchically- or periodically-specified.

## Keywords

Succinct Specifications Computational Complexity Efficient and Non-Efficient Approximability.## Preview

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