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 in Discret Math 310(3):614–621, 2010). Chalermsook and Chuzhoy (In: Proceedings of the 21st annual ACM-SIAM symposium on discrete algorithms, SODA 2010, Austin, Texas, USA, 17–19 Jan 2010, SIAM, pp 1334–1349, 2010) presented a Linear Programming (LP) 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 et al. (ACM Trans Algorithms 15(2):20:1–20:33, 2019) 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 $\epsilon >0$, and $\mathcal {I}$ be an instance of RMFC where the optimum number of firefighters to save all the leaves is $OPT(\mathcal {I})$. We present an algorithm which uses at most $\lceil (1+\epsilon )OPT(\mathcal {I})\rceil$ many firefighters at each time step and runs in time $n^{O(\log \log n/\epsilon )}$. 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 refined height reduction lemma than the one in Adjiashvili et al. (2019) 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. (2019) plus a more careful analysis to improve their 12-approximation and provide a polynomial time ($5+\epsilon$)-approximation.