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Determinant sums for undirected Hamiltonicity

Björklund, Andreas LU (2010) 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS 2010) In 2010 IEEE 51st Annual Symposium On Foundations Of Computer Science p.173-182
Abstract (Swedish)
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... (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)
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author
organization
publishing date
type
Chapter in Book/Report/Conference proceeding
publication status
published
subject
in
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)
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
2010-09-03 12:04:14
date last changed
2018-05-29 11:29:19
@inproceedings{f47a1070-db82-4c37-bd86-f519303fdde7,
  abstract     = {<b>Abstract in Undetermined</b><br/><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>
<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>
<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},
  year         = {2010},
}