Considering fairness has become increasingly important in recent research. This paper proposes the prize-collecting vertex cover problem with fairness constraints (FPCVC). In a prize-collecting vertex cover problem, those edges that are not covered incur penalties. By adding fairness concerns into the problem, the vertex set is divided into l groups, the goal is to find a vertex set to minimize the cost-plus-penalty value under the constraints that the profit of edges collected by each group exceeds a coverage requirement. In this paper, we propose a hybrid algorithm (combining deterministic rounding and randomized rounding) for the FPCVC problem which, with probability at least \(1-1/l^{\alpha }\), returns a feasible solution with an objective value at most \(\left( \frac{9(\alpha +1)}{2}\ln l+3\right) \) times that of an optimal solution, where \(\alpha \) is a constant. We also show a lower bound of \(\Omega (\ln l)\) for the approximability of FPCVC. Thus, our approximation ratio is asymptotically best possible. Experiments show that our algorithm performs fairly well empirically.

在最近的研究中，考虑公平性变得越来越重要。本文提出了带有公平约束的领奖顶点覆盖问题（FPCVC）。在有奖顶点覆盖问题中，那些未被覆盖的边会受到惩罚。通过在问题中加入公平性问题，将顶点集分为l组，目标是在每组收集的边的利润超过覆盖要求的约束下，找到一个使成本加罚值最小的顶点集。在本文中，我们针对 FPCVC 问题提出了一种混合算法（结合确定性舍入和随机舍入），该算法以至少\(1-1/l^{\alpha }\)的概率返回具有目标值的可行解至多\(\left( \frac{9(\alpha +1)}{2}\ln l+3\right) \)倍于最优解，其中\(\alpha \)是常数。我们还展示了 FPCVC 近似性的下界\(\Omega (\ln l)\) 。因此，我们的近似比率是渐近最好的。实验表明，我们的算法根据经验表现得相当好。