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An efficient swarm evolution algorithm with probability learning for the black and white coloring problem.

Zhiqiang Zhang1,2, Li Zhang3, Xiujun Zhang4

  • 1Key Laboratory of Digital Innovation of Tianfu Culture, Sichuan Provincial Department of Culture and Tourism, Chengdu University, Chengdu, 610106, China. zqzhang@cdu.edu.cn.

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|July 12, 2025
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Summary
This summary is machine-generated.

A new swarm evolution algorithm effectively solves the NP-complete black and white coloring (BWC) problem. This method improves upon existing heuristic algorithms and enhances known best results for BWC challenges.

Keywords:
Black and white coloringEvolutionaryImproved simulated annealingProbability learningSwarm evolution algorithm

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Area of Science:

  • Graph Theory
  • Combinatorial Optimization
  • Computational Complexity

Background:

  • The black and white coloring (BWC) problem involves partitioning graph vertices into two sets, ensuring no adjacent vertices share the same color.
  • It is an NP-complete problem with applications in chemical reagent storage and chess problems.
  • Finding optimal solutions is computationally challenging.

Purpose of the Study:

  • To develop an efficient algorithm for solving the NP-complete black and white coloring (BWC) problem.
  • To maximize the number of white vertices in a graph coloring scheme with a fixed number of black vertices.
  • To improve upon existing heuristic algorithms for BWC.

Main Methods:

  • A novel swarm evolution algorithm is proposed, integrating improved simulated annealing search.
  • Key algorithmic components include a probability learning mechanism, crossover, perturbation, and tabu search.
  • The algorithm's performance is evaluated on various graph types, including Cayley, random, semi-random, and DIMACS benchmark graphs.

Main Results:

  • The proposed swarm evolution algorithm demonstrates superior performance compared to classical heuristic algorithms for the BWC problem.
  • The algorithm successfully improved upon previously known best results for the BWC problem.
  • Experimental results validate the effectiveness and efficiency of the developed swarm evolution approach.

Conclusions:

  • The swarm evolution algorithm offers a powerful and effective method for tackling the black and white coloring problem.
  • This approach provides a significant advancement in solving NP-complete graph coloring problems.
  • The algorithm's ability to enhance existing solutions highlights its potential for practical applications.