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.
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Nagarajan, V., Ravi, R. (2008). The Directed Minimum Latency Problem. In: Goel, A., Jansen, K., Rolim, J.D.P., Rubinfeld, R. (eds) Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2008 2008. Lecture Notes in Computer Science, vol 5171. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85363-3_16
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DOI: https://doi.org/10.1007/978-3-540-85363-3_16
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