Asymptotic Quasi-Polynomial Time Approximation Scheme for Resource Minimization for Fire Containment
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.
Firefighter Problem
Resource Management
Fire Containment
Approximation Algorithm
Asymptotic Approximation Scheme
Theory of computation
33:1-33:14
Regular Paper
Authors supported by organization NSERC.
Mirmahdi
Rahgoshay
Mirmahdi Rahgoshay
Department of Computing Science, University of Alberta, Edmonton, Alberta, T6G 2E8, Canada
Mohammad R.
Salavatipour
Mohammad R. Salavatipour
Department of Computing Science, University of Alberta, Edmonton, Alberta, T6G 2E8, Canada
10.4230/LIPIcs.STACS.2020.33
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Mirmahdi Rahgoshay and Mohammad R. Salavatipour
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