Logarithmic Approximations for Fair k-Set Selection

We study the fair k-set selection problem where we aim to select k sets from a given set system such that the (weighted) occurrence times that each element appears in these k selected sets are balanced, i.e., the maximum (weighted) occurrence times are minimized. By observing that a set system can be formulated into a bipartite graph G := (L ∪ R, E), our problem is equivalent to selecting k vertices from R such that the maximum (weighted) number selected neighbors of vertices in L is minimized. The problem arises in a wide range of applications in various fields, such as machine learning, artificial intelligence, and operations research. We first prove that the problem is NP-hard even if the maximum degree ∆ of the input bipartite graph is 3, and the problem is in P when ∆ = 2. We then show that the problem is also in P when the input set system forms a laminar family. Based on intuitive linear programming, we show that two rounding algorithms achieve (Equation presented)-approximation on general bipartite graphs, and an independent rounding algorithm achieves O(log ∆)-approximation on bipartite graphs with a maximum degree ∆. We demonstrate that our analysis is almost tight by providing a hard instance for this linear programming.