Open questions in the theory of semifeasible computation
ACM SIGACT News
Theoretical Computer Science
Logspace algorithms for computing shortest and longest paths in series-parallel graphs
FSTTCS'07 Proceedings of the 27th international conference on Foundations of software technology and theoretical computer science
Logspace optimization problems and their approximability properties
FCT'05 Proceedings of the 15th international conference on Fundamentals of Computation Theory
FCT'07 Proceedings of the 16th international conference on Fundamentals of Computation Theory
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We study the problem of finding a path between two vertices in finite directed graphs whose independence number is bounded by some constant k. The independence number of a graph is the largest number of vertices that can be picked such that there is no edge between any two of them. The complexity of this problem depends on the exact question we ask: Do we wish only to tell whether a path exists? Do we also wish to construct such a path? Are we required to construct the shortest one? Concerning the first question, we show that the reachability problem is first-order definable for all k and that its succinct version is $\Pi_2^{\mathrm{P}}$-complete for all k. In contrast, the reachability problems for many other types of finite graphs, including dags and trees, are not first-order definable, and their succinct versions are PSPACE-complete. Concerning the second question, we show not only that we can construct paths in logarithmic space, but that there even exists a logspace approximation scheme for this problem. The scheme gets a ratio r 1 as additional input and outputs a path that is at most r times as long as the shortest path. Concerning the third question, we show that even telling whether the shortest path has a certain length is NL-complete and thus is as difficult as for arbitrary directed graphs.