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The K r -Packing Problem

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Abstract

For a fixed integer r≥2, the K r -packing problem is to find the maximum number of pairwise vertex-disjointK r 's (complete graphs on r vertices) in a given graph. The K r -factor problem asks for the existence of a partition of the vertex set of a graph into K r 's. The K r -packing problem is a natural generalization of the classical matching problem, but turns out to be much harder for r≥3 – it is known that for r≥3 the K r -factor problem is NP-complete for graphs with clique number r [16]. This paper considers the complexity of the K r -packing problem on restricted classes of graphs.

We first prove that for r≥3 the K r -packing problem is NP-complete even when restrict to chordal graphs, planar graphs (for r=3, 4 only), line graphs and total graphs. The hardness result for K 3-packing on chordal graphs answers an open question raised in [6]. We also give (simple) polynomial algorithms for the K 3-packing and the K r -factor problems on split graphs (this is interesting in light of the fact that K r -packing becomes NP-complete on split graphs for r≥4), and for the K r -packing problem on cographs.

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Authors and Affiliations

  1. MIT Laboratory for Computer Science Cambridge, MA 01239, USA venkat@theory.lcs.mit.edu, , , , , , US

    Venkatesan Guruswami

  2. Department of Computer Science and Engineering Indian Institute of Technology Madras-600 036, India rangan@iitm.ernet.in, , , , , , IN

    C. Pandu Rangan

  3. Department of Computer Science and Information Engineering National Chung Cheng University Ming-Hsiun, Chiayi 621 Taiwan, Republic of China mschang@cs.ccu.edu.tw, , , , , , TW

    M. S. Chang

  4. Department of Applied Mathematics National Chaio Tung University Hsinchu 30050 Taiwan, Republic of China gjchang@math.nctu.edu.tw, , , , , , TW

    G. J. Chang

  5. Department of Computer Science and Engineering Chinese University of Hong Kong, Hong Kong wongck@cse.cuhk.edu.hk, , , , , , CN

    C. K. Wong

Authors
  1. Venkatesan Guruswami
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  2. C. Pandu Rangan
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  3. M. S. Chang
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  4. G. J. Chang
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  5. C. K. Wong
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Additional information

Received September 27, 1999; revised August 14, 2000

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Guruswami, V., Pandu Rangan, C., Chang, M. et al. The K r -Packing Problem. Computing 66, 79–89 (2001). https://doi.org/10.1007/s006070170039

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  • DOI: https://doi.org/10.1007/s006070170039

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