New Integrality Gap Results for the Firefighters Problem on Trees
In the firefighter problem on trees, we are given a tree \(G=(V,E)\) together with a vertex \(s \in V\) where the fire starts spreading. At each time step, the firefighters can pick one vertex while the fire spreads from burning vertices to all their neighbors that have not been picked. The process stops when the fire can no longer spread. The objective is to find a strategy that maximizes the total number of vertices that do not burn. This is a simple mathematical model, introduced in 1995, that abstracts the spreading nature of, for instance, fire, viruses, and ideas. The firefighter problem is NP-hard and admits a \((1-1/e)\) approximation via LP rounding. Recently, a PTAS was announced in .(The \((1-1/e)\) approximation remained the best until very recently when Adjiashvili et al.  showed a PTAS. Their PTAS does not bound the LP gap.)
The goal of this paper is to develop better understanding on the power of LP relaxations for the firefighter problem. We first show a matching lower bound of \((1-1/e+\epsilon )\) on the integrality gap of the canonical LP. This result relies on a powerful combinatorial gadget that can be used to derive integrality gap results in other related settings. Next, we consider the canonical LP augmented with simple additional constraints (as suggested by Hartke). We provide several evidences that these constraints improve the integrality gap of the canonical LP: (i) Extreme points of the new LP are integral for some known tractable instances and (ii) A natural family of instances that are bad for the canonical LP admits an improved approximation algorithm via the new LP. We conclude by presenting a 5 / 6 integrality gap instance for the new LP.
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