Fighting constrained fires in graphs

Bonato, Anthony; Messinger, Margaret-Ellen; Praat, Pawe

doi:10.1016/j.tcs.2012.01.041

Cited by 8 publications

(8 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the Firefighter problem, one firefighter cannot contain a fire on the infinite Cartesian grid [17]. It was shown in [15] that if an average of 3 2 + 1 3x+2 vertices are protected at each step (protecting at most 2 vertices at each step), the fire can be contained by time step 12x + 7 for x ∈ N. A similar result was independently proven in [16]; they additionally proved that any ratio greater than 3 2 can be achieved. In both [15] and [16], the authors also conjectured that if firefighters protect an average of exactly 3 2 vertices at each step, a fire on an infinite Cartesian grid cannot be contained; this was later proven by [5].…”

Section: The Infinite Cartesian Gridmentioning

confidence: 99%

“…We then consider the Pyro game on the infinite Cartesian and strong grids. In the Firefighter problem, one firefighter cannot contain a fire on the infinite Cartesian grid, but an average of slightly more than 3 2 firefighters can. In Section 3, we show that in the Pyro game, one firefighter can contain a fire on the infinite Cartesian grid.…”

Section: Introduction To the Pyro Gamementioning

confidence: 99%

See 1 more Smart Citation

The Pyro game: a slow intelligent fire

Messinger,

Yarnell

2021

Preprint

Self Cite

View full text Add to dashboard Cite

In the Firefighter problem, a fire breaks out at a vertex of a graph and at each subsequent time step, the firefighter chooses a vertex to protect and then the fire spreads from each burned vertex to every unprotected neighbour. The problem can be thought of as a simplified model for the spread of gossip or disease in a network. We introduce a new two-player variation called the Pyro game, in which at each step, the fire spreads from one burned vertex to all unprotected neighbours of that vertex. The fire is no longer automated and aims to maximize the number of burned vertices. We show, that unlike the Firefighter problem, one firefighter can contain a fire on the Cartesian grid in the Pyro game. We also study both the Pyro Game and the Firefighter Problem on the infinite strong grid and the complexity of the Pyro game.

show abstract

Section: The Infinite Cartesian Gridmentioning

confidence: 99%

Section: Introduction To the Pyro Gamementioning

confidence: 99%

The Pyro game: a slow intelligent fire

Messinger,

Yarnell

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Any progress in this direction tells us how lack of information impacts the quality of the solution. Note that a version of the game introduced in [7] also models a lack of information. In that version, rather than the firefighting resources, the missing information is where the fire will spread.…”

Section: Our Contributionmentioning

confidence: 99%

“…, we obtain Equation (7). Now, Equations (5) and (7) imply that in case (ii), when y = b, we also have…”

mentioning

confidence: 99%

Firefighting on trees

Coupechoux

Demange

Ellison

et al. 2019

Theoretical Computer Science

View full text Add to dashboard Cite

“…The study of this concept has become very fruitful in the literature and the evidence is the existence of many works on the subject for different graph structures [9,6,14,16,29,36,41,44,42,43].…”

Section: Introductionmentioning

confidence: 99%

Firefighting as a Strategic Game

Àlvarez¹,

Blesa²,

Molter³

2015

Internet Mathematics

View full text Add to dashboard Cite

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.

show abstract

Fighting constrained fires in graphs

Cited by 8 publications

References 11 publications

The Pyro game: a slow intelligent fire

The Pyro game: a slow intelligent fire

Firefighting on trees

Firefighting as a Strategic Game

Contact Info

Product

Resources

About