Firefighting on trees

Coupechoux, Pierre; Demange, Marc; Ellison, David; Jouve, Bertrand

doi:10.1016/j.tcs.2019.01.040

Cited by 3 publications

(2 citation statements)

References 16 publications

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“…[15,16] considered the Max-Save Firefighter problem and gave polynomial time algorithm for interval graphs, split graphs, permutation graphs, and proved NP-hardness of the problem on Unit Disk Graphs. Online firefighter problem and fractional firefighter problem were introduced by [17] and they also gave the optimal competitive ratio and asymptotic comparison of number of firefighters and the size of the tree levels. NP-completeness of different graph problems on unit disk graphs have been discussed by [18].…”

Section: Previous Workmentioning

confidence: 99%

Firefighter Problem with Minimum Budget: Hardness and Approximation Algorithm for Unit Disk Graphs

Chatterjee¹,

Bhattacharyya²

2022

Preprint

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Unit disk graphs are the set of graphs which represent the intersection of disk graphs and interval graphs. These graphs are of great importance due to their structural similarity with wireless communication networks. Firefighter problem on unit disk graph is interesting as it models the virus spreading in an wireless network and asks for a solution to stop it. In this paper, we consider the MIN-BUDGET firefighter problem where the goal is to determine the minimum number of firefighters required and the nodes to place them at each time instant to save a given set of vertices of a given graph and a fire breakout node. We show that, the MIN-BUDGET firefighter problem in a unit disk graph is NP-Hard. We also present a constant factor approximation algorithm.

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Section: Previous Workmentioning

confidence: 99%

Firefighter Problem with Minimum Budget: Hardness and Approximation Algorithm for Unit Disk Graphs

Chatterjee¹,

Bhattacharyya²

2022

Preprint

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“…This problem was also discussed on static graphs and with a model in which 1 firefighter is to be placed in each time-step. The firefighting problem is NP-Complete [18,20,30] and has been studied several times in the literature [3,2,8,11,12,13,14,19,25].…”

Section: Related Workmentioning

confidence: 99%

Spread and defend infection in graphs

Gupta¹

2021

Preprint

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The spread of an infection, a contagion, meme, emotion, message and various other spreadable objects have been discussed in several works. Burning and firefighting have been discussed in particular on static graphs. Burning simulates the notion of the spread of "fire" throughout a graph (plus, one unburned node burned at each time-step), graph firefighting simulates the defending of nodes by placing firefighters on the nodes which have not been already burned while the fire is being spread (started by a single fire source).This article studies a combination of firefighting and burning on a graph class which is a variation (generalization) of temporal graphs. Nodes can be infected from "outside" a network. We present a notion of both upgrading (of unburned nodes, similar to firefighting) and repairing (of infected nodes). The nodes which are burned, firefighted, or repaired are chosen probabilistically. So a variable amount of nodes are infected, upgraded and repaired in each time step.In the model presented in this article, both burning and firefighting proceed concurrently, we introduce such a system to enable the community to study the notion of spread of an infection and the notion of upgrade/repair against each other. The graph class that we study these processes is a variation of temporal graph class in which at each timestep, probabilistically, a communication takes place (iff an edge exists in that time step). In addition, a node can be "worn out" and thus can be removed from the network, a new healthy node can be added to the network. This class of graphs enables systems with high complexity to be able to be simulated and studied.

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The Moving Firefighter Problem

et al. 2022

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The original formulation of the firefighter problem defines a discrete-time process where a fire starts at a designated subset of the vertices of a graph G. At each subsequent discrete time unit, the fire propagates from each burnt vertex to all of its neighbors unless they are defended by a firefighter that can move between any pair of vertices in a single time unit. Once a vertex is burnt or defended, it remains in that state, and the process terminates when the fire can no longer spread. In this work, we present the moving firefighter problem, which is a generalization of the firefighter problem where the time it takes a firefighter to move from a vertex u to defend vertex v is determined by a function τ. This new formulation models situations such as a wildfire or a flood, where firefighters have to physically move from their current position to the location of an entity they intend to defend. It also incorporates the notion that entities modeled by the vertices are not necessarily instantaneously defended upon the arrival of a firefighter. We present a mixed-integer quadratically constrained program (MIQCP) for the optimization version of the moving firefighter problem that minimizes the number of burnt vertices for the case of general finite graphs, an arbitrary set F⊂V of vertices where the fire breaks out, a single firefighter, and metric time functions τ.

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Firefighting on trees

Cited by 3 publications

References 16 publications

Firefighter Problem with Minimum Budget: Hardness and Approximation Algorithm for Unit Disk Graphs

Firefighter Problem with Minimum Budget: Hardness and Approximation Algorithm for Unit Disk Graphs

Spread and defend infection in graphs

The Moving Firefighter Problem

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