Minimum Membership Hitting Sets of Axis Parallel Segments

Abstract

The Minimum Membership Set Cover (MMSC) problem is a well studied variant among set covering problems. We study the dual of MMSC problem which we refer to as Minimum Membership Hitting Set (MMHS) problem. Exact Hitting Set (EHS) problem is a special case of MMHS problem. In this paper, we show that EHS problem for hypergraphs induced by horizontal axis parallel segments intersected by vertical axis parallel segments is \(\mathsf {NP}\)-complete. Our reduction shows that finding a hitting set in which the number of times any set is hit is minimized does not admit a \(2-\epsilon \) approximation. In the case when the horizontal segments are intersected by vertical lines (instead of vertical segments), we give an algorithm to optimally solve the MMHS problem in polynomial time. Clearly, this algorithm solves the EHS problem as well. Yet, we present a combinatorial algorithm for the special case of EHS problem for horizontal segments intersected by vertical lines because it provides interesting pointers to forbidden structures of intervals that have exact hitting sets. We also present partial results on such forbidden structures.