A Constant-Factor Approximation for Directed Latency in Quasi-Polynomial Time
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We give the first constant-factor approximation for the Directed Latency problem in quasi-polynomial time. Here, the goal is to visit all nodes in an asymmetric metric with a single vehicle starting at a depot r to minimize the average time a node waits to be visited by the vehicle. The approximation guarantee is an improvement over the polynomial-time O(log n)-approximation [Friggstad, Salavatipour, Svitkina, 2013] and no better quasi-polynomial time approximation algorithm was known. To obtain this, we must extend a recent result showing the integrality gap of the Asymmetric TSP-Path LP relaxation is bounded by a constant [Köhne, Traub, and Vygen, 2019], which itself builds on the breakthrough result that the integrality gap for standard Asymmetric TSP is also a constant [Svensson, Tarnawsi, and Vegh, 2018]. We show the standard Asymmetric TSP-Path integrality gap is bounded by a constant even if the cut requirements of the LP relaxation are relaxed from x(δ^in(S)) ≥ 1 to x(δ^in(S)) ≥ρ for some constant 1/2 < ρ≤ 1. We also give a better approximation guarantee in the special case of Directed Latency in regret metrics where the goal is to find a path P minimize the average time a node v waits in excess of c_rv, i.e. 1/|V|·∑_v ∈ V (c_v(P)-c_rv).
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