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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2020.33
URN: urn:nbn:de:0030-drops-118946
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/11894/
Rahgoshay, Mirmahdi ;
Salavatipour, Mohammad R.
Asymptotic Quasi-Polynomial Time Approximation Scheme for Resource Minimization for Fire Containment
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}
}
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Keywords: |
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Firefighter Problem, Resource Management, Fire Containment, Approximation Algorithm, Asymptotic Approximation Scheme |
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Collection: |
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37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020) |
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Issue Date: |
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2020 |
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Date of publication: |
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04.03.2020 |
