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 rv portion of P. We give the first constantfactor approximation guarantee for DirLat, but in quasipolynomial time. Previously, a polynomialtime O(log n)approximation was known [Z. Friggstad et al., 2013], and no better approximation guarantees were known even in quasipolynomial 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 HeldKarp relaxation for asymmetric TSPPath (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 HeldKarp 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 totalregret problem, where the goal is to find a path P that minimizes the total time that nodes spend in excess of their shortestpath distances from r, which can be cast as a special case of DirLat involving socalled regret metrics.
BibTeX  Entry
@InProceedings{friggstad_et_al:LIPIcs:2020:12918,
author = {Zachary Friggstad and Chaitanya Swamy},
title = {{A ConstantFactor Approximation for Directed Latency in QuasiPolynomial Time}},
booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)},
pages = {52:152:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771627},
ISSN = {18688969},
year = {2020},
volume = {173},
editor = {Fabrizio Grandoni and Grzegorz Herman and Peter Sanders},
publisher = {Schloss DagstuhlLeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/12918},
URN = {urn:nbn:de:0030drops129183},
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 