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Solution cluster structure analysis of random regular 3-satisfiability problems
Lichao PANG, Xiaofeng WANG, Zhixin XIE, Yi YANG, Xingyu ZHAO, Lan YANG
Journal of Computer Applications    2024, 44 (7): 2137-2143.   DOI: 10.11772/j.issn.1001-9081.2023070940
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Regular 3-SATisfiability (3-SAT) problem is an NP-hard problem. Studying alterations in the cluster structure of solutions to regular 3-SAT problem is to enhance the comprehension of the difficulty involved in problem determination and distribution of satisfiable solutions. However, existing analysis models only study a few discrete values near the cluster phase transition point. Under different constraint densities, there is lack of unified analysis model to describe the structural evolution of solution clusters. To solve this problem, a Phase transition Model of Solution cluster Structure (PMSS) was proposed. The main idea of this model is to obtain an initial solution of regular 3-SAT problem using WalkSAT algorithm and information propagation algorithm, construct a solution cluster of initial solutions by using random walks, and analyze the solution cluster. Modularity and community were used to measure community structure, and structural entropy was used to measure the complexity of the solution cluster structure. Experimental results show that PMSS can accurately analyze the structural evolution process of solution clusters, and the phase transition point of regular 3-SAT problem instances is between 13 and 14, which is consistent with the phase transition point obtained using Zchaff solver, further verifying the effectiveness of PMSS.

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