# A Faster Tree-Decomposition Based Algorithm for Counting Linear Extensions

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http://hdl.handle.net/10138/321138

#### Citation

Kangas , K , Koivisto , M & Salonen , S 2019 , ' A Faster Tree-Decomposition Based Algorithm for Counting Linear Extensions ' , Algorithmica , vol. 82 , pp. 2156-2173 . https://doi.org/10.1007/s00453-019-00633-1

 Title: A Faster Tree-Decomposition Based Algorithm for Counting Linear Extensions Author: Kangas, Kustaa; Koivisto, Mikko; Salonen, Sami Contributor: University of Helsinki, Aalto UniversityUniversity of Helsinki, Department of Computer ScienceUniversity of Helsinki, Department of Computer Science Date: 2019-10-03 Language: eng Number of pages: 18 Belongs to series: Algorithmica ISSN: 1432-0541 URI: http://hdl.handle.net/10138/321138 Abstract: We investigate the problem of computing the number of linear extensions of a given n-element poset whose cover graph has treewidth t. We present an algorithm that runs in time $${\tilde{O}}(n^{t+3})$$O~(nt+3)for any constant t; the notation $${\tilde{O}}$$O~hides polylogarithmic factors. Our algorithm applies dynamic programming along a tree decomposition of the cover graph; the join nodes of the tree decomposition are handled by fast multiplication of multivariate polynomials. 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 these parameters leaves large variance in running times due to uncontrolled features of the selected optimal-width tree decomposition. We compare two approaches to select an efficient tree decomposition: one is to include additional features of the tree decomposition to build a more accurate, heuristic cost function; the other approach is to fit a statistical regression model to collected running time data. Both approaches are shown to yield a tree decomposition that typically is significantly more efficient than a random optimal-width tree decomposition. Subject: Algorithm selection COMPLEXITY Empirical hardness FRAMEWORK Linear extension Multiplication of polynomials Tree decomposition 113 Computer and information sciences 111 Mathematics Rights:
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