The Directed Minimum Latency Problem

• Authors: Viswanath Nagarajan;R. Ravi
• Venue: APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
• Year: 2008

We study the directed minimum latency problem: given an n-vertex asymmetric metric (V,d) with a root vertex r茂戮驴 V, find a spanning path originating at rthat minimizes the sum of latencies at all vertices (the latency of any vertex v茂戮驴 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/茂戮驴)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 茂戮驴 0) a polynomial time O(n1/2 + 茂戮驴)-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).