Approximation algorithm for prize-collecting vertex cover with fairness constraints

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 \(1-1/l^{\alpha }\) $1-1/l^{\alpha }$ , returns a feasible solution with an objective value at most \(\left( \frac{9(\alpha +1)}{2}\ln l+3\right) \) $\left(\frac{9(\alpha +1)}{2},ln,l,+,3\right)$ times that of an optimal solution, where \(\alpha \) $\alpha$ is a constant. We also show a lower bound of \(\Omega (\ln l)\) $\mathrm{\Omega }(lnl)$ for the approximability of FPCVC. Thus, our approximation ratio is asymptotically best possible. Experiments show that our algorithm performs fairly well empirically.