Graph Burning: Mathematical Formulations and Optimal Solutions

Abstract

The graph burning problem is an NP-hard combinatorial optimization problem that helps quantify how vulnerable a graph is to contagion. This paper introduces three mathematical formulations of the problem: an integer linear program (ILP) and two constraint satisfaction problems (CSP1 and CSP2). Thanks to off-the-shelf optimization software, these formulations can be solved optimally over arbitrary graphs; this is relevant because the only algorithms designed to date for this problem are approximation algorithms and heuristics, which do not guarantee to find optimal solutions. We empirically compared the proposed formulations using random graphs and off-the-shelf optimization software. The results show that CSP1 and CSP2 tend to reach optimal solutions in less time than the ILP. Therefore, we executed them over some benchmark graphs of order at most 5908. The previously best-known solutions for some of these graphs were improved. We draw some empirical observations from the experimental results. For instance, we find the tendency: the larger the graph’s optimal solution, the more difficult it is to find it. Finally, the resulting set of optimal solutions might be helpful as a benchmark dataset for the performance evaluation of non-exact algorithms.

Random Walks-Based Node Centralities to Attack Complex Networks

“…CLS: While the closeness nodes attack (CLS) performs poorly on most road maps, it is particularly effective on UK Faculty, Little Rock Food Web, and Arenas Email regarding real-world networks. Regarding synthetic networks, it exhibits a fairly good performance overall, especially on ER, BBT, and LTC (20,20). The CLS ranks seventh among the examined strategies, with an R −1 avg ≈ 0.68.…”

“…Among real-world networks, it ranks first on Beijing 3 rd and performs well on San Joaquin County (R −1 ≈ 0.91) and Beijing 4 th (R −1 ≈ 0.95). Among synthetic networks, KSH proves effective on lattices, ranking first on LTC (20,20) and having an R −1 ≈ 0.95 on LTC (20,5). KSH ranks fourth among all tested strategies, with an R −1 avg ≈ 0.80.…”

“…CT: The covering time (CT) proved effective on LTC (20,20), and two real-world networks, Beijing 3 rd and 4 th . It also exhibited noteworthy effectiveness on Beijing 2 nd and ER.…”

“…In other terms, the CT selects nodes covering the whole network structure, and for this, removing nodes according to the CT strategy may cause a faster LCC dismantling. (20,5). Bottom row, the twenty most central nodes according to each node removal strategy for the LTC (20,20).…”

“…The graph burning problem (GBP) is introduced in the context of social contagion, and it may also model the spread of viral infections under a very idealistic context [19,20]. GBP furnishes the burning number, which quantifies how vulnerable to "contagion" a network is.…”