The directed minimum latency problem

We study the directed minimum latency problem: given an n-vertex asymmetric metric (V, d) with a root vertex r ∈ V, find a spanning path originating at r that minimizes the sum of latencies at all vertices (the latency of any vertex v ∈ V is the distance from r to v along the path). This problem has been well-studied on symmetric metrics, and the best known approximation guarantee is 3.59 [3]. For any 1logn < < 1, we give an n O(1/) time algorithm for directed latency that achieves an approximation ratio of O(ρ · n3), where ρ is the integrality gap of an LP relaxation for the asymmetric traveling salesman path problem [13, 5]. We prove an upper bound ρ = O( n), which implies (for any fixed > 0) a polynomial time O(n1/2+)-approximation algorithm for directed latency. In the special case of metrics induced by shortest-paths in an unweighted directed graph, we give an O(log2 n) approximation algorithm. As a consequence, we also obtain an O(log2 n) approximation algorithm for minimizing the weighted completion time in no-wait permutation flowshop scheduling. We note that even in unweighted directed graphs, the directed latency problem is at least as hard to approximate as the well-studied asymmetric traveling salesman problem, for which the best known approximation guarantee is O(log n).