A greedy heuristic for graph burning (2401.07577v3)

Abstract: 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.

• The paper presents a deterministic greedy heuristic that reduces the graph burning problem to a series of clustered maximum coverage challenges.
• It rigorously evaluates the heuristic on networks with up to 102,400 vertices, confirming efficient performance on complex real-world topologies.
• The study highlights potential applications in influence maximization and network resilience, encouraging exploration of parallel processing to further enhance efficiency.

Insights into Graph Burning and Greedy Heuristics

The paper entitled "A Greedy Heuristic for Graph Burning" presents a comprehensive paper on solving the optimization version of the graph burning problem (GBP). This problem is essentially a mathematical model employed to represent processes such as social contagion and information dissemination across networks. The result of the research provides insights into the development of a deterministic greedy heuristic, tailored to address the challenges posed by GBP.

Graph burning is characterized by the identification of a sequence of vertices such that every vertex is within a certain maximum distance from a 'burned' vertex over discrete time steps. This problem essentially measures a graph's vulnerability to contagion, with the core objective being to minimize the length of the burning sequence, known as the burning number $b(G)$.

Key Contributions and Algorithmic Development

The authors highlight the inherent NP-hardness of GBP and its connections to several classic NP-hard problems such as the minimum dominating set, vertex k-center, and the firefighter problem. The significant reduction of this problem to a series of clustered maximum coverage problems (CMCP) stands out, allowing for GBP to be tackled through a series of maximum coverage problems.

The introduction of a deterministic greedy heuristic offers practitioners an avenue for efficiently finding feasible solutions to GBP, especially on large graphs. This heuristic is premised on a simple yet effective approach—leveraging a greedy approximation algorithm for CMCP—to iteratively select subsets that maximize new coverage. This approach efficiently balances the coverage while minimizing the overlapping of already burned vertices.

Performance and Experimental Evaluation

In terms of performance, the heuristic is tested extensively across a variety of real-world and synthetic graphs. Empirical results suggest that, despite theoretical limitations on specific graph structures like paths and cycles, the heuristic performs robustly on complex real-network topologies where it identifies most of the optimal and best-known solutions efficiently.

Particularly impressive is the algorithm's ability to tackle large graphs with up to 102,400 vertices. The paper demonstrates the scalability of the heuristic against real-world networks where its execution time remains feasible. Users could integrate this heuristic as a foundational tool for practical applications in socio-information networks or systems requiring rapid dissemination of information.

Furthermore, the heuristic's variance, referred to as GrP, which repeats the process across different initial configurations, provides increased chances of reaching optimal configurations, though at a higher computational cost. This extension illustrates the potential trade-offs between computational resources and solution quality, an important consideration for practitioners dealing with extensive graph instances.

Implications for Future Research

There are multiple implications for both practical undertakings and theoretical exploration. Practically, the proposed algorithms could find applications in fields requiring efficient influence maximization or problem domains related to network resilience against failures. Theoretically, the reduction from GBP to CMCP and the subsequent heuristic make a case for more in-depth analysis of greedy solutions in complex networks.

Moreover, the relationship between various graph parameters and the efficacy of these heuristics could be a fruitful area for future exploration. Subsequent research might extend this work by refining approximations or exploring heuristic modifications that incorporate parallel processing to further optimize runtime on dense or dynamically evolving network graphs.

In conclusion, this paper effectively demonstrates how a well-developed heuristic approach, grounded in a deep understanding of problem structure, can significantly enhance our ability to manage complex network phenomena like graph burning. Future research building upon these findings has the potential to further unlock efficient solutions across broader classes of graph-theoretic problems.