Parameterized complexity of firefighting

Bazgan, Cristina; Chopin, Morgan; Cygan, Marek; Fellows, Michael R.; Fomin, Fedor V.; Leeuwen, Erik Jan van

doi:10.1016/j.jcss.2014.03.001

Cited by 20 publications

(38 citation statements)

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“…It can be rightfully concluded that the EGM(1, 1) and REGM(1, 1) models are suitable for making forecasts with improved accuracy as compared to extant models. The further work of this study will design some new algorithms for the selection and optimization of industrial forecasting applications (Bazgan et al [37]; Göös et al [38]; Wan et al [39]). …”

Section: Resultsmentioning

confidence: 99%

Dynamic grey platform for efficient forecasting management

Tsai

2015

Journal of Computer and System Sciences

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Section: Resultsmentioning

confidence: 99%

Dynamic grey platform for efficient forecasting management

Tsai

2015

Journal of Computer and System Sciences

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“…The discrete mathematics community pays particularly high attention to the firefighter problem on grids [16,10], and there has also been some work on infinite graphs [13]. The problem also received a lot of attention from the parameterized complexity perspectives [8,3,5] and on many special cases, e.g., when the tree has bounded pathwidth [8] and on bounded degree graphs [8,4].…”

Section: Our Contributionsmentioning

confidence: 99%

New Integrality Gap Results for the Firefighters Problem on Trees

Chalermsook

Vaz

2017

Approximation and Online Algorithms

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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.

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“…It was shown to be fixed parametertractable with respect to the combined parameter "pathwidth" and "maximum degree" [10]. Other important results can be found in [12,3]. From this perspective, the problem is known to be fixed-parameter tractable on trees in various parameterizations.…”

Section: Introductionmentioning

confidence: 99%

Firefighting as a Strategic Game

Àlvarez¹,

Blesa²,

Molter³

2015

Internet Mathematics

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The Firefighter Problem was proposed in 1995 [25] as a deterministic discrete-time model for the spread and containment of a fire. The problem is defined on an undirected finite graph G = (V, E), where initially fire breaks out at f nodes. In each subsequent time-step, two actions occur: A certain number b of firefighters are placed on nonburning nodes, permanently protecting them from the fire. Then the fire spreads to all non-defended neighbors of the nodes on fire. Since the graph is finite, at some point each node is either on fire or saved, and thus the fire cannot spread further. One of the objectives for the problem is to place the firefighters in such a way that the number of saved nodes is maximized.The applications of the Firefighter problem 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. Most research on the problem considers the case in which the fire starts in a single place (i.e., f = 1), and in which the budget of available firefighters per time-step is one (i.e., b = 1). So does the work in this paper too. This configuration already leads to hard problems and even in this case the problem is known to be NP-hard. * This work was supported by grants TIN2013-46181-C2-1-R and TIN2012-37930-C02-2 of the Spanish Ministry of Science and Innovation, and by the project SGR 2014-1034 of the Generalitat de Catalunya.† A preliminary version of this work was presented in 11th Workshop on Algorithms and Models for the Web Graph (waw'14), Beijing, China, December 2014. 1In this work, we study the problem from a game-theoretical perspective. We introduce a strategic game model for the Firefighter Problem to tackle its complexity from a different angle. We refer to it as the Firefighter Game. Such a game-based context seems very appropriate when applied to large networks, where entities may act and make decisions based on their own interests, without global coordination.At every time-step of the game, a player decides whether to place a new firefighter in a non-burning node of the graph. If so, he must decide where to place it. By placing it, the player is indirectly deciding which nodes to protect at that time-step. We define different utility functions in order to model selfish and non-selfish scenarios, which lead to equivalent games. We show that the Price of Anarchy (PoA) is linear for a particular family of graphs, but it is at most 2 for trees. We also analyze the quality of the equilibria when coalitions among players are allowed. It turns out that it is possible to compute an equilibrium in polynomial time, even for constant size coalitions. This yields to a polynomial time approximation algorithm for the problem and its approximation ratio equals the PoA of the corresponding game. We show that for some specific topologies, the PoA is constant when constant size coalitions are considered.

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Parameterized complexity of firefighting

Cited by 20 publications

References 10 publications

Dynamic grey platform for efficient forecasting management

Dynamic grey platform for efficient forecasting management

New Integrality Gap Results for the Firefighters Problem on Trees

Firefighting as a Strategic Game

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