Approximation algorithms for the partition set cover problem with penalties

Abstract

In this paper, we consider the partition set cover problem with penalties. In this problem, we have a universe U, a partition \(\mathscr {P}=\{P_{1},\ldots ,P_{r}\}\) of U, and a collection \(\mathscr {S}=\{S_{1},\ldots ,S_{m}\}\) of nonempty subsets of U satisfying \(\bigcup _{S_i\in \mathscr {S}} S_i=U\). In addition, each \(P_t\) \((t\in [r])\) is associated with a covering requirement \(k_t\) as well as a penalty \(\pi _t\), and each \(S_i\) \((i\in [m])\) is associated with a cost. A class \(P_t\) attains its covering requirement by a subcollection \(\mathscr {A}\) of \(\mathscr {S}\) if at least \(k_t\) elements in \(P_t\) are contained in \(\bigcup _{S_i\in \mathscr {A}} S_i\). Each \(P_t\) is either attaining its covering requirement or paid with its penalty. The objective is to find a subcollection \(\mathscr {A}\) of \(\mathscr {S}\) such that the sum of the cost of \(\mathscr {A}\) and the penalties of classes not attaining covering requirements by \(\mathscr {A}\) is minimized. We present two approximation algorithms for this problem. The first is based on the LP-rounding technique with approximation ratio \(K+O(\beta +\ln r)\), where \(K=\max _{t\in [r]}k_t\), and \(\beta \) denotes the approximation guarantee for a related set cover instance obtained by rounding the standard LP. The second is based on the primal-dual method with approximation ratio lf, where \(f=\max _{e\in U}|\{S_i\in \mathscr {S}\mid e\in S_i\}|\) and \(l=\max _{t\in [r]}|P_t|\).