A Faster Tree-Decomposition Based Algorithm for Counting Linear Extensions

Authors Kustaa Kangas, Mikko Koivisto, Sami Salonen

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Kustaa Kangas
  • Department of Computer Science, Aalto University, Espoo, Finland
Mikko Koivisto
  • Department of Computer Science, University of Helsinki, Helsinki, Finland
Sami Salonen
  • Department of Computer Science, University of Helsinki, Helsinki, Finland

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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)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algorithm design techniques
  • Theory of computation → Parameterized complexity and exact algorithms
  • Algorithm selection
  • empirical hardness
  • linear extension
  • multiplication of polynomials
  • tree decomposition


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