An Integer Programming Formulation for the Maximum k-Subset Intersection Problem

Abstract

In this paper, we study the maximum $k$-subset intersection (M$k$SI) problem. Given an integer $k$, a ground set $U$ and a collection $\mathcal {S}$ of subsets of $U$, the M$k$SI problem is to select $k$ subsets $S_1, S_2, \ldots , S_k$ in $\mathcal {S}$ whose intersection size $|S_1 \cap S_2 \cap \dots \cap S_k|$ is maximum. The M$k$SI problem is NP-hard and hard to approximate. Some applications of the M$k$SI problem can be found in the literature and, to the best of our knowledge, no exact method was proposed to solve this problem. In this work, we introduce a very effective preprocessing procedure to reduce the size of the input, introduce a GRASP heuristic which was able to find solutions very close to be optimal ones, propose an integer programming formulation for the problem and present computational experiments made with instances that come from an application.