## 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~ (n^{t} ^{+} ^{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.

Original language | English |
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Pages (from-to) | 2156-2173 |

Number of pages | 18 |

Journal | Algorithmica |

Volume | 82 |

Issue number | 8 |

DOIs | |

Publication status | Published - 1 Aug 2020 |

MoE publication type | A1 Journal article-refereed |

## Keywords

- Algorithm selection
- Empirical hardness
- Linear extension
- Multiplication of polynomials
- Tree decomposition