The Directed Minimum Latency Problem

Abstract

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 $\frac{1}{\log n}<\epsilon<1$, we give an n ^O(1/ε) time algorithm for directed latency that achieves an approximation ratio of $O(\rho\cdot \frac{n^\epsilon}{\epsilon^3})$, where ρ is the integrality gap of an LP relaxation for the asymmetric traveling salesman path problem [13,5]. We prove an upper bound $\rho=O(\sqrt{n})$, which implies (for any fixed ε> 0) a polynomial time O(n ^1/2 + ε)-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(logn).

Supported by NSF grant CCF-0728841.