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

2025-12-18

AI Summary

This paper studies the Resource Minimization for Fire Containment (RMFC) problem and the Non-Uniform k-Center (NUkC) problem on trees. RMFC seeks the minimum budget $B$—the number of vertices that can be protected per time step—to prevent fire from reaching a designated subset of critical vertices; NUkC aims to achieve near-optimal clustering using the fewest possible additional centers beyond $k$. The authors introduce novel techniques including LP-guided enumeration, double scaling, and combinatorial structural analysis. They obtain the first optimal 2-approximation for RMFC on trees and an asymptotic PTAS, improving the prior best approximation ratio from 3 to the theoretical optimum. Moreover, they establish a deep connection between RMFC and NUkC, yielding the first optimal approximation algorithm for NUkC in terms of the number of extra centers—a long-standing open problem in the area.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Applications
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Security and Privacy: Large-scale security measurements

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.

Problem

• Designs optimal approximation for fire containment resource minimization on trees
• Develops asymptotic PTAS for smooth variant using LP-guided enumeration
• Extends techniques to non-uniform k-center for optimal additional centers

Innovation

• PTAS for smooth RMFC via LP-guided enumeration
• Optimal 2-approximation for Resource Minimization Fire Containment
• First optimal approximation for non-uniform k-center problem