Solving the Graph Burning Problem for Large Graphs

TL;DR

This work tackles the Graph Burning Problem, an NP-hard plot of contagion spread on networks, by introducing PRYM, an exact algorithm based on an integer-programming formulation with row generation. PRYM solves GBP decision problems via binary search on the burning length $B$ and uses on-demand loading of covering constraints, aided by a distance-on-demand variant of the BFF heuristic (BFF-d) to obtain a feasible start. Empirically, PRYM delivers substantial improvements over the prior best, GDCA, solving real-world graphs up to about $2\times 10^5$ vertices in under 35 seconds and achieving an average speedup of around $236$x on smaller instances, while requiring only a small fraction of the full covering constraints. The results extend the frontier of GBP solvability to much larger networks and point to future work in column generation and branch-and-price to further scale the approach on even larger graphs.

Abstract

We propose an exact algorithm for the Graph Burning Problem ($\texttt{GBP}$), an NP-hard optimization problem that models the spread of influence on social networks. Given a graph $G$ with vertex set $V$, the objective is to find a sequence of $k$ vertices in $V$, namely, $v_1, v_2, \dots, v_k$, such that $k$ is minimum and $\bigcup_{i = 1}^{k} \{u\! \in\! V\! : d(u, v_i) \leq k - i\} = V$, where $d(u,v)$ denotes the distance between $u$ and $v$. We formulate the problem as a set covering integer programming model and design a row generation algorithm for the $\texttt{GBP}$. Our method exploits the fact that a very small number of covering constraints is often sufficient for solving the integer model, allowing the corresponding rows to be generated on demand. To date, the most efficient exact algorithm for the $\texttt{GBP}$, denoted here by $\texttt{GDCA}$, is able to obtain optimal solutions for graphs with up to 14,000 vertices within two hours of execution. In comparison, our algorithm finds provably optimal solutions approximately 236 times faster, on average, than $\texttt{GDCA}$. For larger graphs, memory space becomes a limiting factor for $\texttt{GDCA}$. Our algorithm, however, solves real-world instances with almost 200,000 vertices in less than 35 seconds, increasing the size of graphs for which optimal solutions are known by a factor of 14.

Figures (1)

• Figure 1: Illustration of the vertices (colored and enlarged) whose covering constraints suffice to prove the infeasibility of GBP-IP for $B = b(G) - 1$.

Theorems & Definitions (6)

• Theorem 1
• Conjecture 1
• Proposition 1
• proof
• Proposition 2
• proof