奖励-收集顶点覆盖问题的精确算法

Exact algorithm for the prize-collecting vertex covering problem

奖励-收集顶点覆盖问题是顶点覆盖问题的衍生问题,同时也是组合优化NP-hard问题。本文提出该问题的数学性质并给出证明,利用数学性质能够确定某些顶点一定在或一定不在最优奖励-收集顶点覆盖集中,从而降低该问题的规模;基于该问题的数学性质设计出上下界子算法、降阶子算法、回溯子算法,通过降阶子算法可以降低该问题的规模,从而缩短回溯子算法的搜索时间,进而降低求解该问题最优解的时间。通过应用和算法对比表明,所设计的算法比没有考虑该问题数学性质的一般精确算法的时间复杂度更低。

The prize-collecting vertex covering problem is not only a derivative of the vertex covering problem,but also a problem for NP-hard combinatorial optimization.In this paper,we firstly propose the mathematical properties of the problem and give the proof,which can determine that some vertices must or must not be in the optimal prize collection vertex covering set,so as to reduce the scale of the problem.Secondly,based on the mathematical properties of the problem,we design the upper and lower bound sub-algorithm,reduced-order sub-algorithm and backtracking sub-algorithm.Through the reduced-order sub-algorithm,the scale of the problem can be reduced,so as to shorten the search time of the backtracking sub-algorithm,and then reduce the time to solve the optimal solution of the problem.A comparison of applications and algorithms shows that the designed algorithm has lower time complexity than the general accurate algorithms without considering the mathematical properties of the problem.