Logspace Optimization Problems and Their Approximability Properties

Abstract

This paper introduces logspace optimization problems as analogues of the well-studied polynomial-time optimization problems. Similarly to them, logspace optimization problems can have vastly different approximation properties, even though the underlying decision problems have the same computational complexity. Natural problems, including the shortest path problems for directed graphs, undirected graphs, tournaments, and forests, exhibit such a varying complexity. In order to study the approximability of logspace optimization problems in a systematic way, polynomial-time approximation classes are transferred to logarithmic space. Appropriate reductions are defined and optimization problems are presented that are complete for these classes. It is shown that under the assumption L ≠ NL some logspace optimization problems cannot be approximated with a constant ratio; some can be approximated with a constant ratio, but do not permit a logspace approximation scheme; and some have a logspace approximation scheme, but cannot be solved in logarithmic space. A new natural NL-complete problem is presented that has a logspace approximation scheme.