The Firefighter Problem: A Structural Analysis

We consider the complexity of the firefighter problem where b>=1 firefighters are available at each time step. This problem is proved NP-complete even on trees of degree at most three and budget one (Finbow et al.,2007) and on trees of bounded degree b+3 for any fixed budget b>=2 (Bazgan et al.,2012). In this paper, we provide further insight into the complexity landscape of the problem by showing that the pathwidth and the maximum degree of the input graph govern its complexity. More precisely, we first prove that the problem is NP-complete even on trees of pathwidth at most three for any fixed budget b>=1. We then show that the problem turns out to be fixed parameter-tractable with respect to the combined parameter "pathwidth" and "maximum degree" of the input graph

New Integrality Gap Results for the Firefighters Problem on Trees

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.Comment: 22 page

Firefighting as a game

The Firefighter Problem was proposed in 1995 [16] as a deterministic discrete-time model for the spread (and containment) of a fire. Its applications reach from real fires to the spreading of diseases and the containment of floods. Furthermore, it can be used to model the spread of computer viruses or viral marketing in communication networks. In this work, we study the problem from a game-theoretical perspective. Such a context seems very appropriate when applied to large networks, where entities may act and make decisions based on their own interests, without global coordination. We model the Firefighter Problem as a strategic game where there is one player for each time step who decides where to place the firefighters. We show that the Price of Anarchy is linear in the general case, but at most 2 for trees. We prove that the quality of the equilibria improves when allowing coalitional cooperation among players. In general, we have that the Price of Anarchy is in T(n/k) where k is the coalition size. Furthermore, we show that there are topologies which have a constant Price of Anarchy even when constant sized coalitions are considered.Peer ReviewedPostprint (author’s final draft

Approximation algorithms for network design and cut problems in bounded-treewidth

This thesis explores two optimization problems, the group Steiner tree and firefighter problems, which are known to be NP-hard even on trees. We study the approximability of these problems on trees and bounded-treewidth graphs. In the group Steiner tree, the input is a graph and sets of vertices called groups; the goal is to choose one representative from each group and connect all the representatives with minimum cost. We show an O(log^2 n)-approximation algorithm for bounded-treewidth graphs, matching the known lower bound for trees, and improving the best possible result using previous techniques. We also show improved approximation results for group Steiner forest, directed Steiner forest, and a fault-tolerant version of group Steiner tree. In the firefighter problem, we are given a graph and a vertex which is burning. At each time step, we can protect one vertex that is not burning; fire then spreads to all unprotected neighbors of burning vertices. The goal is to maximize the number of vertices that the fire does not reach. On trees, a classic (1-1/e)-approximation algorithm is known via LP rounding. We prove that the integrality gap of the LP matches this approximation, and show significant evidence that additional constraints may improve its integrality gap. On bounded-treewidth graphs, we show that it is NP-hard to find a subpolynomial approximation even on graphs of treewidth 5. We complement this result with an O(1)-approximation on outerplanar graphs.Diese Arbeit untersucht zwei Optimierungsprobleme, von welchen wir wissen, dass sie selbst in Bäumen NP-schwer sind. Wir analysieren Approximationen für diese Probleme in Bäumen und Graphen mit begrenzter Baumweite. Im Gruppensteinerbaumproblem, sind ein Graph und Mengen von Knoten (Gruppen) gegeben; das Ziel ist es, einen Knoten von jeder Gruppe mit minimalen Kosten zu verbinden. Wir beschreiben einen O(log^2 n)-Approximationsalgorithmus für Graphen mit beschränkter Baumweite, dies entspricht der zuvor bekannten unteren Schranke für Bäume und ist zudem eine Verbesserung über die bestmöglichen Resultate die auf anderen Techniken beruhen. Darüber hinaus zeigen wir verbesserte Approximationsresultate für andere Gruppensteinerprobleme. Im Feuerwehrproblem sind ein Graph zusammen mit einem brennenden Knoten gegeben. In jedem Zeitschritt können wir einen Knoten der noch nicht brennt auswählen und diesen vor dem Feuer beschützen. Das Feuer breitet sich anschließend zu allen Nachbarn aus. Das Ziel ist es die Anzahl der Knoten die vom Feuer unberührt bleiben zu maximieren. In Bäumen existiert ein lang bekannter (1-1/e)-Approximationsalgorithmus der auf LP Rundung basiert. Wir zeigen, dass die Ganzzahligkeitslücke des LP tatsächlich dieser Approximation entspricht, und dass weitere Einschränkungen die Ganzzahligkeitslücke möglicherweise verbessern könnten. Für Graphen mit beschränkter Baumweite zeigen wir, dass es NP-schwer ist, eine sub-polynomielle Approximation zu finden