Graph burning | Encyclopedia MDPI

Graph burning: Comparison

Please note this is a comparison between Version 4 by Jesús García Díaz and Version 3 by Jesús García Díaz.

This e ntry is adapted from the peer-reviewed paper 2227-7390/10/15/2777

The graph burning problem is a relatively new combinatorial optimization problem that helps quantify howa graph's vulnerable a graph is to contagionility. It is defined in terms of a fundamental diffusion model.

graph burning
social contagion

1. Introduction

The graph burning problem (GBP) is an NP-hard combinatorial optimization problem introduced in 2014 in the context of social contagion ^[1]. This problem, concerned with the sequential spread of information over a graph, considers that information can be spread from different places and times ^[1][2]. In this entry, by graph we refer to an undirected simple graph ^[3]. Computer network message propagation is an example of a real-world problem that the GBP may model; in this scenario, an initial spreading entity can send a message one host at a time, while these hosts can propagate the message only to their neighbors ^[4][5]. The GBP may model other real-world problems: social contagion in social networks and the spread of viral infections under a very idealistic context ^[6].

The GBP receives a graph G=(V,E) as input, and its goal is to find a minimum length sequence of vertices

(s_{1}, s_{2}, \dots, s_{k})

that burns all graph’s vertices by following the burning process. This process consists of repeating the following steps from

i = 1

to k, where all vertices are unburned at the beginning, and once a vertex is burned, it remains in that state.

a) The neighbors of the burned vertices get burned.

b) Vertex $s_{i}$ gets burned.

Figure 1 shows a simple example of the burning process. The length of an optimal burning sequence for a given graph G is denoted by b(G) and is known as the burning number.

Figure 1. An optimal burning sequence for a nine vertices path is $(v_{3}, v_{7}, v_{9})$ . The burning process of this sequence is depicted by each $i \in {1, 2, 3}$ with its corresponding steps a and b. Notice that vertex $v_{3}$ burns all vertices at distance at most 2, vertex $v_{7}$ burns all vertices at distance at most 1, and vertex $v_{9}$ only burns itself.

2. Related work

The decision version of the GBP is NP-complete. This problem receives as input a graph

G = (V, E)

and a positive integer k; it asks if a burning sequence of length at most k exists. The problem remains NP-complete when restricted to trees of maximum degree three, chordal graphs, bipartite graphs, planar graphs, spider graphs, and disconnected graphs ^[2]. The optimization version of the problem also remains NP-hard for trees and graphs with disjoint paths ^[7]. Regarding arbitrary graphs, two approximation algorithms are reported in the literature; they have an approximation factor of 3 and

3 - 2 / b (G)

, respectively ^[7][8]. There is a 2-approximation algorithm for trees, a 1.5-approximation algorithm for graphs with disjoint paths, and a 2-approximation algorithm for square grids ^[6][7]. For minimization problems, a

ρ

-approximation algorithm returns solutions of size at most

ρ \cdot O P T

, where

O P T

is the size of the optimal solution and

ρ \geq 1

. In the case of maximization problems, a

ρ

-approximation algorithm returns solutions of size at least

(1 / ρ) \cdot O P T

^[9]. Besides approximation algorithms, some heuristics have been proposed too ^[4][5]; these are mainly based on centrality measures and binary search over the set of possible values of

b (G)

. The GBP has been formulated using integer linear programming and constraint satisfaction problems. Using these formulations and off-the-shelf optimization software, graphs with up to 5908 vertices have been solved to proven optimality ^[10].

The GBP has been approached mostly from a theoretical point of view. As a result, many of its properties over specific graph families have been identified. Among these families are paths ^[2][7], trees ^[2][7], grids ^[6], intervals ^[6], fences ^[11], theta ^[12], spiders ^[13], path-forests ^[2][7][13], caterpillars ^[14], products ^[15], and generalized Petersen graphs ^[16]. Some of the main properties of the GBP are the following. All paths and cycles G of order n have

b (G) = ⌈ n^{1 / 2} ⌉

, all graphs G with a Hamiltonian path have

b (G) \leq ⌈ n^{1 / 2} ⌉

^[1][2], all spiders and caterpillars G have

b (G) \leq ⌈ n^{1 / 2} ⌉

^[13][14], all complete graphs G of order at least two have

b (G) = 2

, and all perfect binary trees G of depth r have

b (G) = r + 1

. Based on these properties, a conjecture on the upper bound of the burning number of connected graphs was formulated by Bonato et al. [1]: Every connected graph G of order n has burning number b(G)

b (G) \leq ⌈ n^{1 / 2} ⌉

. This conjecture, known as the burning number conjecture, is one of the most important open questions in the area. To date, the best-known bound for the burning number of arbitrary connected graphs is

b (G) \leq ⌈ {(4 \cdot n / 3)}^{1 / 2} ⌉ + 1

^[17].

The GBP resembles other NP-hard problems, such as the vertex k-center problem (VKCP) and the firefighter problem (FP). The VKCP consists of finding the best location for a set of

k \in Z^{+}

centers, where such locations are the ones that minimize the maximum distance a customer has to travel to its nearest center ^[18][19][20]. Although the VKCP and the GBP are different, their approximation algorithms are conceptually similar ^[7][8][20]. This similarity comes from the fact that the VKCP has a polynomial-time reduction to the minimum dominating set problem, which can be viewed as the problem of burning all vertices in parallel in one single step ^{[20][21][22][23]}. Regarding the FP, it aims at protecting vertices from burning given an initial set of fire sources ^[24][25][26]. Although GBP and FP have different goals, the latter’s integer linear program (ILP) served as inspiration to define an integer linear program for the GBP ^[10].

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