The Directed Minimum Latency Problem

We study the directed minimum latency problem: given an n-vertex asymmetric metric (V,d) with a root vertex ri¾? V, find a spanning path originating at rthat minimizes the sum of latencies at all vertices (the latency of any vertex vi¾? Vis the distance from rto valong the path). This problem has been well-studied on symmetric metrics, and the best known approximation guarantee is 3.59 [3]. For any $\frac{1}{\log n} , we give an nO(1/i¾?)time algorithm for directed latency that achieves an approximation ratio of $O(\rho\cdot \frac{n^\epsilon}{\epsilon^3})$, where ρis the integrality gap of an LP relaxation for the asymmetric traveling salesman pathproblem [13,5]. We prove an upper bound $\rho=O(\sqrt{n})$, which implies (for any fixed i¾?> 0) a polynomial time O(n1/2 + i¾?)-approximation algorithm for directed latency. In the special case of metrics induced by shortest-paths in an unweighted directed graph, we give an O(log2n) approximation algorithm. As a consequence, we also obtain an O(log2n) 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(logn).