New Integrality Gap Results for the Firefighters Problem on Trees

TL;DR

This paper establishes a tight integrality gap for the canonical LP of the firefighter problem on trees, introduces enhanced LP formulations with better approximation guarantees, and provides new theoretical bounds and instances.

Contribution

It presents the first matching integrality gap for the canonical LP and proposes improved LP relaxations with evidence of better approximation bounds.

Findings

Matching integrality gap of (1-1/e+ε) for the canonical LP.

Enhanced LP with additional constraints improves integrality gap.

A 5/6 integrality gap instance for the new LP.

Abstract

The firefighter problem is NP-hard and admits a $(1-1/e)$ approximation based on rounding the canonical LP. In this paper, we first show a matching integrality gap of $(1-1/e+\epsilon)$ on the canonical LP. This result relies on a powerful combinatorial gadget that can be used to prove integrality gap results for many problem settings. We also 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.