Determinant sums for undirected Hamiltonicity
(2010) 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS 2010) p.173-182- Abstract
- Abstract in Undetermined
We present a Monte Carlo algorithm for Hamiltonicity detection in an n-vertex undirected graph running in O*(1.657(n)) time. To the best of our knowledge, this is the first superpolynomial improvement on the worst case runtime for the problem since the O*(2(n)) bound established for TSP almost fifty years ago (Bellman 1962, Held and Karp 1962). It answers in part the first open problem in Woeginger's 2003 survey on exact algorithms for NP-hard problems.
For bipartite graphs, we improve the bound to O*(1.414(n)) time. Both the bipartite and the general algorithm can be implemented to use space polynomial in n.
We combine several recently resurrected ideas to get the results. Our main technical... (More) - Abstract in Undetermined
We present a Monte Carlo algorithm for Hamiltonicity detection in an n-vertex undirected graph running in O*(1.657(n)) time. To the best of our knowledge, this is the first superpolynomial improvement on the worst case runtime for the problem since the O*(2(n)) bound established for TSP almost fifty years ago (Bellman 1962, Held and Karp 1962). It answers in part the first open problem in Woeginger's 2003 survey on exact algorithms for NP-hard problems.
For bipartite graphs, we improve the bound to O*(1.414(n)) time. Both the bipartite and the general algorithm can be implemented to use space polynomial in n.
We combine several recently resurrected ideas to get the results. Our main technical contribution is a new reduction inspired by the algebraic sieving method for k-Path (Koutis ICALP 2008, Williams IPL 2009). We introduce the Labeled Cycle Cover Sum in which we are set to count weighted arc labeled cycle covers over a finite field of characteristic two. We reduce Hamiltonicity to Labeled Cycle Cover Sum and apply the determinant summation technique for Exact Set Covers (Bjorklund STACS 2010) to evaluate it. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/1666240
- author
- Björklund, Andreas LU
- organization
- publishing date
- 2010
- type
- Chapter in Book/Report/Conference proceeding
- publication status
- published
- subject
- host publication
- 2010 IEEE 51st Annual Symposium On Foundations Of Computer Science
- pages
- 173 - 182
- publisher
- IEEE - Institute of Electrical and Electronics Engineers Inc.
- conference name
- 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS 2010)
- conference location
- Las Vegas, United States
- conference dates
- 2010-10-23 - 2010-10-26
- external identifiers
-
- wos:000287040100019
- scopus:78751486551
- ISSN
- 0272-5428
- ISBN
- 978-0-7695-4244-7
- DOI
- 10.1109/FOCS.2010.24
- project
- Exact algorithms
- language
- English
- LU publication?
- yes
- id
- f47a1070-db82-4c37-bd86-f519303fdde7 (old id 1666240)
- date added to LUP
- 2016-04-01 14:57:37
- date last changed
- 2022-04-22 06:03:47
@inproceedings{f47a1070-db82-4c37-bd86-f519303fdde7, abstract = {{Abstract in Undetermined<br/>We present a Monte Carlo algorithm for Hamiltonicity detection in an n-vertex undirected graph running in O*(1.657(n)) time. To the best of our knowledge, this is the first superpolynomial improvement on the worst case runtime for the problem since the O*(2(n)) bound established for TSP almost fifty years ago (Bellman 1962, Held and Karp 1962). It answers in part the first open problem in Woeginger's 2003 survey on exact algorithms for NP-hard problems.<br/><br/>For bipartite graphs, we improve the bound to O*(1.414(n)) time. Both the bipartite and the general algorithm can be implemented to use space polynomial in n.<br/><br/>We combine several recently resurrected ideas to get the results. Our main technical contribution is a new reduction inspired by the algebraic sieving method for k-Path (Koutis ICALP 2008, Williams IPL 2009). We introduce the Labeled Cycle Cover Sum in which we are set to count weighted arc labeled cycle covers over a finite field of characteristic two. We reduce Hamiltonicity to Labeled Cycle Cover Sum and apply the determinant summation technique for Exact Set Covers (Bjorklund STACS 2010) to evaluate it.}}, author = {{Björklund, Andreas}}, booktitle = {{2010 IEEE 51st Annual Symposium On Foundations Of Computer Science}}, isbn = {{978-0-7695-4244-7}}, issn = {{0272-5428}}, language = {{eng}}, pages = {{173--182}}, publisher = {{IEEE - Institute of Electrical and Electronics Engineers Inc.}}, title = {{Determinant sums for undirected Hamiltonicity}}, url = {{http://dx.doi.org/10.1109/FOCS.2010.24}}, doi = {{10.1109/FOCS.2010.24}}, year = {{2010}}, }