Logspace Optimization Problems and Their Approximability Properties

Logspace optimization problems are the logspace analogues of the well-studied polynomial-time optimization problems. Similarly to them, logspace optimization problems can have vastly different approximation properties even though their 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 and polynomial-time reductions between optimization problems are transferred to logarithmic space. It is proved that natural problems are complete for different logspace approximation classes. This is used to show 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 optimal solutions cannot be computed in logarithmic space.