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Matroid and Knapsack Center Problems

  • Conference paper
Integer Programming and Combinatorial Optimization (IPCO 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7801))

Abstract

In the classic k-center problem, we are given a metric graph, and the objective is to open k nodes as centers such that the maximum distance from any vertex to its closest center is minimized. In this paper, we consider two important generalizations of k-center, the matroid center problem and the knapsack center problem. Both problems are motivated by recent content distribution network applications. Our contributions can be summarized as follows:

  1. 1

    We consider the matroid center problem in which the centers are required to form an independent set of a given matroid. We show this problem is NP-hard even on a line. We present a 3-approximation algorithm for the problem on general metrics. We also consider the outlier version of the problem where a given number of vertices can be excluded as the outliers from the solution. We present a 7-approximation for the outlier version.

  2. 2

    We consider the (multi-)knapsack center problem in which the centers are required to satisfy one (or more) knapsack constraint(s). It is known that the knapsack center problem with a single knapsack constraint admits a 3-approximation. However, when there are at least two knapsack constraints, we show this problem is not approximable at all. To complement the hardness result, we present a polynomial time algorithm that gives a 3-approximate solution such that one knapsack constraint is satisfied and the others may be violated by at most a factor of 1 + ε. We also obtain a 3-approximation for the outlier version that may violate the knapsack constraint by 1 + ε.

This work was supported in part by the National Basic Research Program of China Grant 2011CBA00300, 2011CBA00301, the National Natural Science Foundation of China Grant 61033001, 61061130540, 61073174, 61202009. The research of D.Z. Chen was supported in part by NSF under Grants CCF-0916606 and CCF-1217906.

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Author information

Authors and Affiliations

  1. Department of Computer Science and Engineering, University of Notre Dame, Notre Dame, IN, 46556, USA

    Danny Z. Chen

  2. Institute for Interdisciplinary Information Sciences (IIIS), Tsinghua University, Beijing, 100084, China

    Jian Li & Hongyu Liang

  3. Department of Computer Science, Utah State University, Logan, UT, 84322, USA

    Haitao Wang

Authors
  1. Danny Z. Chen
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  2. Jian Li
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  3. Hongyu Liang
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  4. Haitao Wang
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Editor information

Editors and Affiliations

  1. Massachusetts Institute of Technology, 77 Massachusetts Ave., 02139, Cambridge, MA, USA

    Michel Goemans

  2. Universidad de Chile, Santiago, Chile

    José Correa

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Chen, D.Z., Li, J., Liang, H., Wang, H. (2013). Matroid and Knapsack Center Problems. In: Goemans, M., Correa, J. (eds) Integer Programming and Combinatorial Optimization. IPCO 2013. Lecture Notes in Computer Science, vol 7801. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36694-9_10

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  • DOI: https://doi.org/10.1007/978-3-642-36694-9_10

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-36693-2

  • Online ISBN: 978-3-642-36694-9

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