Efficient Algorithms for Solving the Graph Burning Problem on Large-Scale Networks

Core Concepts

An efficient exact algorithm for finding optimal burning sequences that minimize the time required to propagate influence over an entire social network represented as a graph.

Abstract

The paper proposes an exact algorithm, called PRYM, for solving the Graph Burning Problem (GBP), which models the spread of influence on social networks. The GBP is an NP-hard optimization problem that seeks to find a sequence of vertices (a burning sequence) that minimizes the time required to burn the entire graph. The key contributions of the paper are: PRYM formulates the GBP as an integer programming (IP) model and employs a row generation approach to efficiently solve the problem. This allows PRYM to find provably optimal solutions significantly faster than the previously best known exact algorithm, GDCA. PRYM is able to solve real-world instances with up to 200,000 vertices in less than 35 seconds, increasing the size of graphs for which optimal solutions are known by a factor of 14 compared to GDCA. The row generation approach in PRYM exploits the observation that a very small number of covering constraints (often just dozens) are often sufficient to prove the feasibility or infeasibility of the IP model for a given burning sequence length. This leads to substantial improvements in running time and memory usage compared to previous methods. The paper demonstrates the effectiveness of PRYM through extensive computational experiments on a large benchmark of real-world network instances. The results show that PRYM outperforms the state-of-the-art GDCA algorithm by a significant margin, both in terms of running time and the size of instances that can be solved optimally.

Stats

The burning number of the ia-enron-only graph is 4. The burning number of the DD244 graph is 7. The burning number of the ca-netscience graph is 6.

Quotes

"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." "To date, the most efficient exact algorithm for the GBP, denoted here by 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 GDCA."

Key Insights Distilled From

Solving the Graph Burning Problem for Large Graphs

by Felipe de Ca... at arxiv.org 04-29-2024

https://arxiv.org/pdf/2404.17080.pdf

Deeper Inquiries

How can the row generation approach in PRYM be further improved or extended to solve even larger instances of the Graph Burning Problem

The row generation approach in PRYM can be further improved or extended to solve even larger instances of the Graph Burning Problem by incorporating advanced techniques in constraint generation and problem decomposition. One way to enhance the row generation process is to implement a more efficient constraint separation algorithm that can quickly identify the most critical covering constraints needed to determine the feasibility of the burning sequence. By optimizing the constraint separation procedure, PRYM can reduce the computational burden of loading unnecessary constraints and focus on generating only the most impactful constraints. Moreover, PRYM can benefit from adaptive constraint generation strategies that dynamically adjust the generation of covering constraints based on the current state of the problem. By intelligently selecting which constraints to add to the model during the solving process, PRYM can improve its efficiency and scalability when handling larger instances with hundreds of thousands of vertices. Additionally, exploring parallel computing techniques to distribute the constraint generation and solving tasks across multiple processors or nodes can significantly accelerate the solution process for extremely large graphs. By leveraging parallel computing resources, PRYM can exploit the computational power of modern hardware to tackle complex instances of the Graph Burning Problem efficiently.

What are the limitations of the Graph Burning Problem model, and how could it be extended or adapted to better capture real-world social network dynamics

The Graph Burning Problem model, while effective in capturing the spread of influence on social networks, has certain limitations that could be addressed through extensions or adaptations to better reflect real-world dynamics. One limitation is the assumption of a deterministic burning process, where vertices become burned based on predefined rules. To better capture the stochastic nature of information diffusion in social networks, the model could be extended to incorporate probabilistic burning mechanisms, where the likelihood of a vertex getting burned is influenced by various factors such as node centrality, community structure, and external stimuli. Furthermore, the current model focuses on finding a burning sequence of minimum length to cover the entire graph, but it does not consider the temporal aspects of the spreading process. By introducing time-dependent constraints or incorporating temporal dynamics into the burning process, the model could better simulate the gradual propagation of influence over time in real-world networks. Additionally, the Graph Burning Problem model could be adapted to account for dynamic network structures, where edges and vertices evolve over time. By integrating dynamic graph representations and adaptive burning strategies, the model can better capture the changing nature of social interactions and information flow in dynamic networks.

What other combinatorial optimization problems in network science or graph theory could benefit from a similar row generation-based approach as used in PRYM

Other combinatorial optimization problems in network science or graph theory that could benefit from a similar row generation-based approach as used in PRYM include: Target Set Selection Problem: This problem involves selecting a minimum set of nodes in a network such that all nodes are influenced within a limited number of steps. By applying a row generation algorithm similar to PRYM, the problem can be efficiently solved for large-scale networks with dynamic influence propagation. Virus Spread Minimization: In this problem, the goal is to minimize the spread of a virus or contagion in a network by strategically immunizing a subset of nodes. A row generation approach can be employed to iteratively identify critical nodes for immunization, optimizing the containment of the virus spread. Network Resilience Optimization: This problem focuses on enhancing the resilience of a network against failures or attacks by strategically reinforcing certain nodes or edges. By utilizing a row generation algorithm, the problem can be solved iteratively to identify the most effective resilience strategies for different network structures.