A Logspace Approximation Scheme for the Shortest Path Problem for Graphs with Bounded Independence Number

How difficult is it to find 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 only wish to tell whether a path exists? Do we also wish to construct such a path? Are we required to construct the shortest path? Concerning the first question, it is known that the reachability problem is first-order definable for all k. In contrast, the corresponding reachability problems for many other types of finite graphs, including dags and trees, are not first-order definable. Concerning the second question, in this paper it is shown that not only can we construct paths in logarithmic space, but there even exists a logspace approximation scheme for this problem. It gets an additional input r>1 and outputs a path that is at most r times as long as the shortest path. In contrast, for directed graphs, undirected graphs, and dags we cannot construct paths in logarithmic space (let alone approximate the shortest one), unless complexity class collapses occur. Concerning the third question, it is shown that even telling whether the shortest path has a certain length is NL-complete and thus as difficult as for arbitrary directed graphs.