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.
@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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