### Abstract

An NP-hard graph problem may be intractable for general graphs but it could be efficiently solvable using dynamic programming for graphs with bounded treewidth. Employing dynamic programming on a tree decomposition usually uses exponential space. In 2010, Lokshtanov and Nederlof introduced an elegant framework to avoid exponential space by algebraization. Later, Fürer and Yu modified the framework in a way that even works when the underlying set is dynamic, thus applying it to tree decompositions. In this work, we design space-efficient algorithms to count the number of Hamiltonian cycles and furthermore solve the Traveling Salesman problem, using polynomial space while the time complexity is only slightly increased. This might be inevitable since we are reducing the space usage from an exponential amount (in dynamic programming solutions) to polynomial. We give an algorithm to count the number of Hamiltonian cycles in time (formula presented) using (formula presented) space, where M(r) is the time complexity to multiply two integers, each of which being represented by at most r bits. Then, we solve the more general Traveling Salesman problem in time (formula presented) using space O(Wkdnlog n), where k and d are the width and the depth of the given tree decomposition and W is the sum of weights. Furthermore, this algorithm counts the number of Hamiltonian Cycles.

Original language | English (US) |
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Title of host publication | Computer Science – Theory and Applications - 14th International Computer Science Symposium in Russia, CSR 2019, Proceedings |

Editors | René van Bevern, Gregory Kucherov |

Publisher | Springer Verlag |

Pages | 38-49 |

Number of pages | 12 |

ISBN (Print) | 9783030199548 |

DOIs | |

State | Published - Jan 1 2019 |

Event | 14th International Computer Science Symposium in Russia, CSR 2019 - Novosibirsk, Russian Federation Duration: Jul 1 2019 → Jul 5 2019 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 11532 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 14th International Computer Science Symposium in Russia, CSR 2019 |
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Country | Russian Federation |

City | Novosibirsk |

Period | 7/1/19 → 7/5/19 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Computer Science – Theory and Applications - 14th International Computer Science Symposium in Russia, CSR 2019, Proceedings*(pp. 38-49). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 11532 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-030-19955-5_4