Toward Optimal Approximations for Resource-Minimization for Fire Containment on Trees and Non-Uniform k-Center

TL;DR

This paper presents optimal approximation algorithms for resource minimization in fire containment on trees and extends these techniques to the non-uniform k-center problem, resolving key open questions.

Contribution

It provides the first optimal 2-approximation and asymptotic PTAS for RMFC on trees, and introduces new methods applicable to non-uniform k-center problems.

Findings

Achieved an optimal 2-approximation for RMFC.

Developed an asymptotic PTAS for a smooth variant of RMFC.

First approximation algorithm for NUkC with optimal number of additional centers.

Abstract

One of the most elementary spreading models on graphs can be described by a fire spreading from a burning vertex in discrete time steps. At each step, all neighbors of burning vertices catch fire. A well-studied extension to model fire containment is to allow for fireproofing a number $B$ of non-burning vertices at each step. Interestingly, basic computational questions about this model are computationally hard even on trees. One of the most prominent such examples is Resource Minimization for Fire Containment (RMFC), which asks how small $B$ can be chosen so that a given subset of vertices will never catch fire. Despite recent progress on RMFC on trees, prior work left a significant gap in terms of its approximability. We close this gap by providing an optimal $2$-approximation and an asymptotic PTAS, resolving two open questions in the literature. Both results are obtained in a unified way, by first designing a PTAS for a smooth variant of RMFC, which is obtained through a careful LP-guided enumeration procedure. Moreover, we show that our new techniques, with several additional ingredients, carry over to the non-uniform $k$-center problem (NUkC), by exploiting a link between RMFC on trees and NUkC established by Chakrabarty, Goyal, and Krishnaswamy. This leads to the first approximation algorithm for NUkC that is optimal in terms of the number of additional centers that have to be opened.