Burning Graphs Through Farthest-First Traversal

Jesus Garcia-Diaz,Lil Maria Xibai Rodriguez-Henriquez,Julio Cesar Perez-Sansalvador,Jose Alejandro Cornejo-Acosta

doi:10.1109/access.2022.3159695

Abstract

Graph burning is a process to determine the spreading of information in a graph. If a sequence of vertices burns all the vertices of a graph by following the graph burning process, then such a sequence is known as a burning sequence. The graph burning problem consists in finding a minimum length burning sequence for a given graph. The solution to this NP-hard combinatorial optimization problem helps quantify a graph’s vulnerability to contagion. This paper introduces a simple farthest-first traversal-based approximation algorithm for this problem over arbitrary graphs. We refer to this proposal as the Burning Farthest-First (BFF) algorithm. BFF runs in <inline-formula> <tex-math notation="LaTeX">$O(n^{3})$ </tex-math></inline-formula> steps and has a tight approximation factor of <inline-formula> <tex-math notation="LaTeX">$3-2/b(G)$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$b(G)$ </tex-math></inline-formula> is the size of an optimal solution. The main attribute of BFF is that it has a better approximation factor than the state-of-the-art approximation algorithms for arbitrary graphs, which report an approximation factor of 3. Despite being simple, BFF proved practical when tested over some benchmark datasets.

Highlights

T HE graph burning problem is an NP-hard combinatorial optimization problem introduced in 2014 as a contagion model in social networks [1]
This paper introduces two 3 − 2/b(G)-approximation algorithms (Algorithms 1 and 2) for the graph burning problem over general graphs, where b(G) ∈ Z+ is the size of an optimal solution
Given the complete weighted graph Gw = (V, Ew) from Observation 2, the graph burning problem seeks a minimum length sequence of vertices f ∗ : {x ∈ Z+ | x ≤ b(G)} → S∗ covering the whole set V (Eq 2), where b(G) is its length, S∗ ⊆ V, |S∗| ≤ b(G), f ∗−1(v) is the set of positions that vertex v has in the sequence, and b(G)−min f ∗−1(v) is the biggest covering radius associated with vertex v

Summary

INTRODUCTION

T HE graph burning problem is an NP-hard combinatorial optimization problem introduced in 2014 as a contagion model in social networks [1]. The main attribute of this problem is that it helps quantify how vulnerable a graph is to contagion This problem’s input is a simple graph G = (V, E), and its goal is to find a minimum length sequence of vertices (x1, x2, ..., xk) such that, by repeating the following steps from i = 1 to k, all vertices in V get burned [1], [5], [6]. This paper introduces two 3 − 2/b(G)-approximation algorithms (Algorithms 1 and 2) for the graph burning problem over general graphs, where b(G) ∈ Z+ is the size of an optimal solution. While Algorithm 1 requires the optimal burning sequence length in advance, making it impractical, Algorithm 2 does not have this requirement The latter is the main contribution of this paper.

GRAPH BURNING PROBLEM

PROPOSED APPROXIMATION ALGORITHMS

RUNNING TIME

TIGHT EXAMPLES

A BRIEF EMPIRICAL EVALUATION

CONCLUSIONS