Optimizing Polynomial-Time Solutions to a Network Weighted Vertex Cover Game

Weighted vertex cover(WVC)is one of the most important combinatorial optimization problems.In this paper,we provide a new game optimization to achieve efficiency and time of solutions for the WVC problem of weighted networks.We first model the WVC problem as a general game on weighted networks.Under the framework of a game,we newly define several cover states to describe the WVC problem.Moreover,we reveal the relationship among these cover states of the weighted network and the strict Nash equilibriums(SNEs)of the game.Then,we propose a game-based asynchronous algorithm(GAA),which can theoretically guarantee that all cover states of vertices converging in an SNE with polynomial time.Subsequently,we improve the GAA by adding 2-hop and 3-hop adjustment mechanisms,termed the improved game-based asynchronous algorithm(IGAA),in which we prove that it can obtain a better solution to the WVC problem than using a the GAA.Finally,numerical simulations demonstrate that the proposed IGAA can obtain a better approximate solution in promising computation time compared with the existing representative algorithms.