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A Faster Tree-Decomposition Based Algorithm for Counting Linear Extensions

Authors: Kustaa Kangas, Mikko Koivisto, and Sami Salonen

Published in: LIPIcs, Volume 115, 13th International Symposium on Parameterized and Exact Computation (IPEC 2018)


Abstract
We consider the problem of counting the linear extensions of an n-element poset whose cover graph has treewidth at most t. We show that the problem can be solved in time O~(n^{t+3}), where O~ suppresses logarithmic factors. Our algorithm is based on fast multiplication of multivariate polynomials, and so differs radically from a previous O~(n^{t+4})-time inclusion - exclusion algorithm. We also investigate the algorithm from a practical point of view. We observe that the running time is not well characterized by the parameters n and t alone, fixing of which leaves large variance in running times due to uncontrolled features of the selected optimal-width tree decomposition. For selecting an efficient tree decomposition we adopt the method of empirical hardness models, and show that it typically enables picking a tree decomposition that is significantly more efficient than a random optimal-width tree decomposition.

Cite as

Kustaa Kangas, Mikko Koivisto, and Sami Salonen. A Faster Tree-Decomposition Based Algorithm for Counting Linear Extensions. In 13th International Symposium on Parameterized and Exact Computation (IPEC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 115, pp. 5:1-5:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)


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@InProceedings{kangas_et_al:LIPIcs.IPEC.2018.5,
  author =	{Kangas, Kustaa and Koivisto, Mikko and Salonen, Sami},
  title =	{{A Faster Tree-Decomposition Based Algorithm for Counting Linear Extensions}},
  booktitle =	{13th International Symposium on Parameterized and Exact Computation (IPEC 2018)},
  pages =	{5:1--5:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-084-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{115},
  editor =	{Paul, Christophe and Pilipczuk, Michal},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2018.5},
  URN =		{urn:nbn:de:0030-drops-102062},
  doi =		{10.4230/LIPIcs.IPEC.2018.5},
  annote =	{Keywords: Algorithm selection, empirical hardness, linear extension, multiplication of polynomials, tree decomposition}
}
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