Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH scholarly article en Rahgoshay, Mirmahdi; Salavatipour, Mohammad R. https://www.dagstuhl.de/lipics License: Creative Commons Attribution 3.0 Unported license (CC-BY 3.0)
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URN: urn:nbn:de:0030-drops-118946
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Asymptotic Quasi-Polynomial Time Approximation Scheme for Resource Minimization for Fire Containment

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Abstract

Resource Minimization Fire Containment (RMFC) is a natural model for optimal inhibition of harmful spreading phenomena on a graph. In the RMFC problem on trees, we are given an undirected tree G, and a vertex r where the fire starts at, called root. At each time step, the firefighters can protect up to B vertices of the graph while the fire spreads from burning vertices to all their neighbors that have not been protected so far. The task is to find the smallest B that allows for saving all the leaves of the tree. The problem is hard to approximate up to any factor better than 2 even on trees unless P = NP [King and MacGillivray, 2010]. Chalermsook and Chuzhoy [Chalermsook and Chuzhoy, 2010] presented a Linear Programming based O(log^* n) approximation for RMFC on trees that matches the integrality gap of the natural Linear Programming relaxation. This was recently improved by Adjiashvili, Baggio, and Zenklusen [Adjiashvili et al., 2017] to a 12-approximation through a combination of LP rounding along with several new techniques. In this paper we present an asymptotic QPTAS for RMFC on trees. More specifically, let ε>0, and ℐ be an instance of RMFC where the optimum number of firefighters to save all the leaves is OPT(ℐ). We present an algorithm which uses at most ⌈(1+ε)OPT(ℐ)⌉ many firefighters at each time step and runs in time n^O(log log n/ε). This suggests that the existence of an asymptotic PTAS is plausible especially since the exponent is O(log log n), not O(log n). Our result combines a more powerful height reduction lemma than the one in [Adjiashvili et al., 2017] with LP rounding and dynamic programming to find the solution. We also apply our height reduction lemma to the algorithm provided in [Adjiashvili et al., 2017] plus a more careful analysis to improve their 12-approximation and provide a polynomial time (5+ε)-approximation.

BibTeX - Entry

@InProceedings{rahgoshay_et_al:LIPIcs:2020:11894,
  author =	{Mirmahdi Rahgoshay and Mohammad R. Salavatipour},
  title =	{{Asymptotic Quasi-Polynomial Time Approximation Scheme for Resource Minimization for Fire Containment}},
  booktitle =	{37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
  pages =	{33:1--33:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-140-5},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{154},
  editor =	{Christophe Paul and Markus Bl{\"a}ser},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/11894},
  URN =		{urn:nbn:de:0030-drops-118946},
  doi =		{10.4230/LIPIcs.STACS.2020.33},
  annote =	{Keywords: Firefighter Problem, Resource Management, Fire Containment, Approximation Algorithm, Asymptotic Approximation Scheme}
}

Keywords: Firefighter Problem, Resource Management, Fire Containment, Approximation Algorithm, Asymptotic Approximation Scheme
Seminar: 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)
Issue date: 2020
Date of publication: 04.03.2020


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