New Integrality Gap Results for the Firefighters Problem on Trees

Abstract

In the firefighter problem on trees, we are given a tree G = (V, E) together with a vertex s ∈ 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 based on rounding the canonical LP. Recently, a PTAS was announced [1]. 1 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 + ) 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. 1 The (1 − 1/e) approximation remained the best until very recently when Adjiashvili et al. [1] showed a PTAS. Their PTAS does not bound the LP gap.

Firefighting on Trees Beyond Integrality Gaps

“…Additional directions of research on the Firefighter problem include parameterized complexity (Cai, Verbin and Yang [6], Bazgan, Chopin and Fellows [3], Cygan, Fomin and van Leeuwen [13] and Bazgan, Chopin, Cygan, Fellows, Fomin and van Leeuwen [2]), generalizations to the case of many initial fires and many firefighters (Bazgan, Chopin and Ries [4] and Costa, Dantas, Dourado, Penso and Rautenbach [11]), and the study of potential strengthenings of the canonical LP for the Firefighter problem on trees (Hartke [23] and Chalermsook and Vaz [9]).…”

“…We recall that the integrality gap of LP RMFC was shown to be Θ(log * n) [8], and the integrality gap of LP FF is asymptotically 1 − 1/e (when n → ∞) [9].…”

“…Furthermore, if OPT ∩ Q x = ∅, by applying Theorem 12 to y ∧ χ V >h with µ = 1 and q = 2, a set R ⊆ V >h is obtained such that R ∪ A is a feasible solution to RMFC with respect to the budget 6 · B OPT . 9 Proof. Notice that OPT ∩ Q x = ∅ implies that for each u ∈ Γ \ W x , we either have A ∩ P u = ∅ and thus a vertex of A cuts u off from the root, or the set OPT contains a vertex on P u ∩ V >h .…”

“…Whereas there are still considerable gaps between current hardness results and approximation algorithms for both the Firefighter problem and RMFC on trees, the currently best approximations essentially match the integrality gaps of the underlying LPs. More precisely, Chalermsook and Vaz [9] showed that for any ǫ > 0, the canonical LP used for the Firefighter problem on trees has an integrality gap of 1 − 1 e + ǫ. This generalized a previous result by Cai, Verbin and Yang [6], who showed the same gap if the integral solution is required to lie in the support of an optimal LP solution.…”

Online Firefighting on Trees

“…It was supported by extensive experimental tests [11], but finding such a constant and proving the ratio is one of the open problems proposed in [2] (Problem 7). It was shown in [15] that such a constant must be greater than e e−1 . Theorem 1 can be expressed by λ Gr ≤ β I ≤ β F ≤ 2λ Gr , which shows that ρ = 2 is such a constant: Corollary 1.…”