Firefighter Problem with Minimum Budget: Hardness and Approximation Algorithm for Unit Disk Graphs

TL;DR

This paper studies the firefighter problem on unit disk graphs, proving its NP-hardness and providing a constant-factor approximation algorithm for minimizing firefighters needed to save specific nodes.

Contribution

It establishes the NP-hardness of the MIN-BUDGET firefighter problem on unit disk graphs and introduces a constant-factor approximation algorithm.

Findings

NP-hardness of the problem in unit disk graphs

Development of a constant-factor approximation algorithm

Relevance to wireless network virus containment

Abstract

Unit disk graphs are the set of graphs which represent the intersection of disk graphs and interval graphs. These graphs are of great importance due to their structural similarity with wireless communication networks. Firefighter problem on unit disk graph is interesting as it models the virus spreading in an wireless network and asks for a solution to stop it. In this paper, we consider the MIN-BUDGET firefighter problem where the goal is to determine the minimum number of firefighters required and the nodes to place them at each time instant to save a given set of vertices of a given graph and a fire breakout node. We show that, the MIN-BUDGET firefighter problem in a unit disk graph is NP-Hard. We also present a constant factor approximation algorithm.