Jesús García-Díaz | Searcharxiv

arXiv2025

Graph burning: an overview of mathematical programs

The Graph Burning Problem (GBP) is a combinatorial optimization problem that has gained relevance as a tool for quantifying a graph's vulnerability to contagion. Although it is based on a very simple propagation model, its decision version is NP-complete, and its optimization version is NP-hard. Many of its theoretical properties across different graph families have been thoroughly explored, and numerous interesting variants have been proposed. This paper reports novel mathematical programs for the optimization version of the classical GBP. Among the presented programs are a Mixed-Integer Linear Program (MILP), a Constraint Satisfaction Problem (CSP), two Integer Linear Programs (ILP), and two Quadratic Unconstrained Binary Optimization (QUBO) problems. Most optimization solvers can handle these, being QUBO problems of a capital interest in quantum computing. The primary aim of this paper is to gain a comprehensive understanding of the GBP by examining its different formulations. Compared to other mathematical programs from the literature, the ones presented here are conceptually simpler and involve fewer variables. These make them more practical for finding optimal solutions using optimization algorithms and solvers, as we show by solving some instances with millions of vertices in just a few minutes.

cs.DM↗

arXiv2024

A greedy heuristic for graph burning

Given a graph $G$, the optimization version of the graph burning problem seeks for a sequence of vertices, $(u_1,u_2,...,u_p) \in V(G)^p$, with minimum $p$ and such that every $v \in V(G)$ has distance at most $p-i$ to some vertex $u_i$. The length $p$ of the optimal solution is known as the burning number and is denoted by $b(G)$, an invariant that helps quantify the graph's vulnerability to contagion. This paper explores the advantages and limitations of an $\mathcal{O}(mn + pn^2)$ deterministic greedy heuristic for this problem, where $n$ is the graph's order, $m$ is the graph's size, and $p$ is a guess on $b(G)$. This heuristic is based on the relationship between the graph burning problem and the clustered maximum coverage problem, and despite having limitations on paths and cycles, it found most of the optimal and best-known solutions of benchmark and synthetic graphs with up to 102400 vertices. Beyond practical advantages, our work unveils some of the fundamental aspects of graph burning: its relationship with a generalization of a classical coverage problem and compact integer programs. With this knowledge, better algorithms might be designed in the future.

cs.DM↗