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Minimum Membership Covering and Hitting

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WALCOM: Algorithms and Computation (WALCOM 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11355))

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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.

J.S.B. Mitchell—Partially supported by the National Science Foundation (CCF-1526406) and the US-Israel Binational Science Foundation (project 2016116).

S. Pandit—Partially supported by the Indo-US Science & Technology Forum (IUSSTF) under the SERB Indo-US Postdoctoral Fellowship scheme with grant number 2017/94, Department of Science and Technology, Government of India.

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Authors and Affiliations

  1. Stony Brook University, Stony Brook, NY, USA

    Joseph S. B. Mitchell & Supantha Pandit

Authors
  1. Joseph S. B. Mitchell
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  2. Supantha Pandit
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Corresponding author

Correspondence to Supantha Pandit .

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Editors and Affiliations

  1. Indian Institute of Technology Guwahati, Guwahati, India

    Gautam K. Das

  2. Indian Institute of Technology Guwahati, Guwahati, India

    Partha S. Mandal

  3. Indian Statistical Institute, Kolkata, India

    Krishnendu Mukhopadhyaya

  4. Gunma University, Kiryu, Japan

    Shin-ichi Nakano

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Mitchell, J.S.B., Pandit, S. (2019). Minimum Membership Covering and Hitting. In: Das, G., Mandal, P., Mukhopadhyaya, K., Nakano, Si. (eds) WALCOM: Algorithms and Computation. WALCOM 2019. Lecture Notes in Computer Science(), vol 11355. Springer, Cham. https://doi.org/10.1007/978-3-030-10564-8_31

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  • DOI: https://doi.org/10.1007/978-3-030-10564-8_31

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-10563-1

  • Online ISBN: 978-3-030-10564-8

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