A Recursive Greedy Algorithm for Walks in Directed Graphs

Given an arc-weighted directed graph G=(V,A) and a pair of nodes s,t, we seek to find an s-t walk of length at most B that maximizes some given function f of the set of nodes visited by the walk. The simplest case is when we seek to maximize the number of nodes visited: this is called the orienteering problem. Our main result is a quasi-polynomial time algorithm that yields an O(log OPT) approximation for this problem when f is a given submodular set function. We then extend it to the case when a node v is counted as visited only if the walk reaches v in its time window [R(v), D(v)]. We apply the algorithm to obtain several new results. First, we obtain an O(log OPT) approximation for a substantial generalization of the orienteering problem in which the profit for visiting each node may vary arbitrarily with time. This captures the time window problem considered earlier for which, even in undirected graphs, the best approximation ratio known is O(log^2 OPT). The second application is an O(log^2 k) approximation for the k-TSP problem in directed graphs (satisfying asymmetric triangle inequality). This is the first non-trivial approximation algorithm for this problem. The third application is an O(log^2 k) approximation (in quasi-poly time) for the group Steiner problem in undirected} graphs where k is the number of groups. This improves earlier ratios by a logarithmic factor and almost matches the inapproximability threshold on trees. This connection to group Steiner trees also enables us to prove that the problem we consider is hard to approximate to a ratio better than Omega(log^{1-eps} OPT), even in undirected graphs. Even though our algorithm runs in quasi-poly time, we believe that the implications for the approximability of several basic optimization problems are interesting. We also believe that, as with the greedy algorithm for the directed Steiner tree problem, obtaining a polynomial time algorithm with a poly-logarithmic approximation ratio would be non-trivial.