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DOI: 10.4230/LIPIcs.FSTTCS.2019.14
URN: urn:nbn:de:0030-drops-115761
URL: https://drops.dagstuhl.de/opus/volltexte/2019/11576/
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Fomin, Fedor V. ; Golovach, Petr A. ; Simonov, Kirill

Parameterized k-Clustering: Tractability Island

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Abstract

In k-Clustering we are given a multiset of n vectors X subset Z^d and a nonnegative number D, and we need to decide whether X can be partitioned into k clusters C_1, ..., C_k such that the cost sum_{i=1}^k min_{c_i in R^d} sum_{x in C_i} |x-c_i|_p^p <= D, where |*|_p is the Minkowski (L_p) norm of order p. For p=1, k-Clustering is the well-known k-Median. For p=2, the case of the Euclidean distance, k-Clustering is k-Means. We study k-Clustering from the perspective of parameterized complexity. The problem is known to be NP-hard for k=2 and it is also NP-hard for d=2. It is a long-standing open question, whether the problem is fixed-parameter tractable (FPT) for the combined parameter d+k. In this paper, we focus on the parameterization by D. We complement the known negative results by showing that for p=0 and p=infty, k-Clustering is W1-hard when parameterized by D. Interestingly, the complexity landscape of the problem appears to be more intricate than expected. We discover a tractability island of k-Clustering: for every p in (0,1], k-Clustering is solvable in time 2^O(D log D) (nd)^O(1).

BibTeX - Entry

@InProceedings{fomin_et_al:LIPIcs:2019:11576,
  author =	{Fedor V. Fomin and Petr A. Golovach and Kirill Simonov},
  title =	{{Parameterized k-Clustering: Tractability Island}},
  booktitle =	{39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)},
  pages =	{14:1--14:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-131-3},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{150},
  editor =	{Arkadev Chattopadhyay and Paul Gastin},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2019/11576},
  URN =		{urn:nbn:de:0030-drops-115761},
  doi =		{10.4230/LIPIcs.FSTTCS.2019.14},
  annote =	{Keywords: clustering, parameterized complexity, k-means, k-median}
}

Keywords: clustering, parameterized complexity, k-means, k-median
Seminar: 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)
Issue Date: 2019
Date of publication: 10.12.2019


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