License: Creative Commons Attribution 3.0 Unported license (CC-BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ESA.2020.52
URN: urn:nbn:de:0030-drops-129183
URL: https://drops.dagstuhl.de/opus/volltexte/2020/12918/
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Friggstad, Zachary ; Swamy, Chaitanya

A Constant-Factor Approximation for Directed Latency in Quasi-Polynomial Time

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Abstract

We consider the directed minimum latency problem (DirLat), wherein we seek a path P visiting all points (or clients) in a given asymmetric metric starting at a given root node r, so as to minimize the sum of the client waiting times, where the waiting time of a client v is the length of the r-v portion of P. We give the first constant-factor approximation guarantee for DirLat, but in quasi-polynomial time. Previously, a polynomial-time O(log n)-approximation was known [Z. Friggstad et al., 2013], and no better approximation guarantees were known even in quasi-polynomial time. A key ingredient of our result, and our chief technical contribution, is an extension of a recent result of [A. Köhne et al., 2019] showing that the integrality gap of the natural Held-Karp relaxation for asymmetric TSP-Path (ATSPP) is at most a constant, which itself builds on the breakthrough similar result established for asymmetric TSP (ATSP) by Svensson et al. [O. Svensson et al., 2018]. We show that the integrality gap of the Held-Karp relaxation for ATSPP 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 for the minimum total-regret problem, where the goal is to find a path P that minimizes the total time that nodes spend in excess of their shortest-path distances from r, which can be cast as a special case of DirLat involving so-called regret metrics.

BibTeX - Entry

@InProceedings{friggstad_et_al:LIPIcs:2020:12918,
  author =	{Zachary Friggstad and Chaitanya Swamy},
  title =	{{A Constant-Factor Approximation for Directed Latency in Quasi-Polynomial Time}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{52:1--52:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-162-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{173},
  editor =	{Fabrizio Grandoni and Grzegorz Herman and Peter Sanders},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12918},
  URN =		{urn:nbn:de:0030-drops-129183},
  doi =		{10.4230/LIPIcs.ESA.2020.52},
  annote =	{Keywords: Approximation Algorithms, Directed Latency, TSP}
}

Keywords: Approximation Algorithms, Directed Latency, TSP
Collection: 28th Annual European Symposium on Algorithms (ESA 2020)
Issue Date: 2020
Date of publication: 26.08.2020


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