# Minimum Membership Covering and Hitting

## Abstract

Set cover is a well-studied problem with application in many fields. A well-known variation of this problem is the Minimum Membership Set Cover problem. In this problem, given a set of points and a set of objects, the objective is to cover all points while minimizing the maximum number of objects that contain any one point. A dual of this problem is the Minimum Membership Hitting Set problem. In this problem, given a set of points and a set of objects, the objective is to stab all of the objects while minimizing the maximum number of points that an object contains. We study both of these variations in a geometric setting with various types of geometric objects in the plane, including axis-parallel line segments, axis-parallel strips, rectangles that are anchored on a horizontal line from one side, rectangles that are stabbed by a horizontal line, and rectangles that are anchored on one of two horizontal lines (i.e., each rectangle shares at least one boundary edge (top or bottom) with one of the input horizontal lines). For each of these problems either we prove NP-hardness or design a polynomial-time algorithm. More precisely, we show that it is NP-complete to decide whether there exists a solution with depth exactly 1 for either the Minimum Membership Set Cover or the Minimum Membership Hitting Set problem. We also provide approximation algorithms for some of the problems. In addition, we study a generalized version of the Minimum Membership Hitting Set problem.

## Keywords

Minimum Membership Set Cover Minimum Membership Hitting Set Rectangles NP-hard Segments Strips Depth of a point## References

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