Game-Based Memetic Algorithm to the Vertex Cover of Networks

The minimum vertex cover (MVC) is a well-known combinatorial optimization problem. A game-based memetic algorithm (GMA-MVC) is provided, in which the local search is an asynchronous updating snowdrift game and the global search is an evolutionary algorithm (EA). The game-based local search can implement (<italic>k,l</italic>)-exchanges for various numbers of <inline-formula> <tex-math notation="LaTeX">${k}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">${l}$ </tex-math></inline-formula> to remove <inline-formula> <tex-math notation="LaTeX">${k}$ </tex-math></inline-formula> vertices from and add <inline-formula> <tex-math notation="LaTeX">${l}$ </tex-math></inline-formula> vertices into the solution set, thus is much better than the previous (1,0)-exchange. Beyond that, the proposed local search is able to deal with the constraint, such that the crossover operator can be very simple and efficient. Degree-based initialization method is also provided which is much better than the previous uniform random initialization. Each individual of the GMA-MVC is designed as a snowdrift game state of the network. Each vertex is treated as an intelligent agent playing the snowdrift game with its neighbors, which is the local refinement process. The game is designed such that its strict Nash equilibrium (SNE) is always a vertex cover of the network. Most of the SNEs are only local optima of the problem. Then an EA is employed to guide the game to escape from those local optimal Nash equilibriums to reach a better Nash equilibrium. From comparison with the state of the art algorithms in experiments on various networks, the proposed algorithm always obtains the best solutions.