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 12approximation 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 12approximation 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 QuasiPolynomial Time Approximation Scheme for Resource Minimization for Fire Containment}},
booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
pages = {33:133:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771405},
ISSN = {18688969},
year = {2020},
volume = {154},
editor = {Christophe Paul and Markus Bl{\"a}ser},
publisher = {Schloss DagstuhlLeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/11894},
URN = {urn:nbn:de:0030drops118946},
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 
Collection: 

37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020) 
Issue Date: 

2020 
Date of publication: 

04.03.2020 