A Faster Tree-Decomposition Based Algorithm for Counting Linear Extensions

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Journal Title
Journal ISSN
Volume Title
A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
Date
2020-08-01
Major/Subject
Mcode
Degree programme
Language
en
Pages
Series
Algorithmica
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 O~ (nt + 3) for any constant t; the notation 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.
Description
Keywords
Algorithm selection, Empirical hardness, Linear extension, Multiplication of polynomials, Tree decomposition
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Citation
Kangas , K , Koivisto , M & Salonen , S 2020 , ' A Faster Tree-Decomposition Based Algorithm for Counting Linear Extensions ' , Algorithmica , vol. 82 , no. 8 , pp. 2156-2173 . https://doi.org/10.1007/s00453-019-00633-1