Graph burning: an overview of mathematical programs

TL;DR

This paper studies the Graph Burning Problem (GBP), which seeks a minimum-length burning sequence to propagate contagion across a graph, with the burning number denoted by $b(G)$. It introduces a family of mathematical programs—MILP, CSP, ILP, and QUBO variants—that encode GBP via the CMCP framework, emphasizing formulations with fewer variables for practical solving. Among these, GBP-ILP stands out for solving large graphs without a binary search, while sQUBO and uQUBO render GBP amenable to quantum/classical hybrid approaches, with uQUBO offering substantial search-space reductions at the cost of careful penalization tuning. The work demonstrates that modern solvers, including row-generation strategies, can handle graphs with millions of vertices within minutes, providing a versatile toolkit for GBP and related contagion- and propagation-models. The results illuminate the practical viability of optimization-based GBP analysis and point to promising future avenues in hybrid classical–quantum methods and GBP variants. $b(G)$ remains a central target for both theory and computation, and this suite of formulations broadens the toolkit for exploring GBP across diverse graph classes and 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.

Figures (9)

• Figure 1: The propagation process of the optimal burning sequence $(v_1,v_5,v_3)$ for the graph $P_5$. At every discrete step, vertices get burned if they are in the neighborhood of a previously burned vertex or if they are in the burning sequence at the current step. Notice that $b(P_5)=3$
• Figure 2: The propagation process of the optimal burning sequence $(v_3,v_2,v_3)$ for the graph $P_5$. Notice that a burning sequence may have repeated vertices and a vertex might have multiple fire sources. For instance, $v_3$ catches fire from $v_3$ and $v_2$
• Figure 3: The CMCP corresponding to $P_4$ with $g=b(P_4)=2$
• Figure 4: An optimal solution, $(v_3,v_2,v_3)$, to PROP-MILP. The input graph is $P_5$ (see Figure \ref{['fig:2']}), $\textbf{S}=[s_{i,j}]_{n \times U}$, $\textbf{B}=[b_{i,j}]_{n \times U}$, and $U=5$. Notice how $\textbf{B}$ correctly codifies the propagation process
• Figure 5: Although matrix $\textbf{B}'$ does not correctly codify the burning process, it is as valid as $\textbf{B}$ from the previous figure

Theorems & Definitions (11)

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