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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},