Graph burning: an overview of mathematical programs

TL;DR

This paper introduces new, simpler mathematical programming formulations for the Graph Burning Problem, enhancing practical solvability and understanding of the problem's complexity and properties.

Contribution

It presents novel MILP, CSP, ILP, and QUBO formulations for GBP, making large instances more tractable and accessible for optimization methods.

Findings

New formulations are simpler and involve fewer variables.

Capable of solving large instances with millions of vertices.

QUBO formulations are suitable for quantum computing applications.

Abstract

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.