Set partitioning via inclusion-exclusion
(2009) In SIAM Journal on Computing 39(2). p.546-563- Abstract
- Given a set N with n elements and a family F of subsets, we show how to partition N into k such subsets in 2(n) n(O)(1) time. We also consider variations of this problem where the subsets may overlap or are weighted, and we solve the decision, counting, summation, and optimization versions of these problems. Our algorithms are based on the principle of inclusion-exclusion and the zeta transform. In effect we get exact algorithms in 2(n) n(O)(1) time for several well-studied partition problems including domatic number, chromatic number, maximum k-cut, bin packing, list coloring, and the chromatic polynomial. We also have applications to Bayesian learning with decision graphs and to model-based data clustering. If only polynomial space is... (More)
- Given a set N with n elements and a family F of subsets, we show how to partition N into k such subsets in 2(n) n(O)(1) time. We also consider variations of this problem where the subsets may overlap or are weighted, and we solve the decision, counting, summation, and optimization versions of these problems. Our algorithms are based on the principle of inclusion-exclusion and the zeta transform. In effect we get exact algorithms in 2(n) n(O)(1) time for several well-studied partition problems including domatic number, chromatic number, maximum k-cut, bin packing, list coloring, and the chromatic polynomial. We also have applications to Bayesian learning with decision graphs and to model-based data clustering. If only polynomial space is available, our algorithms run in time 3(n) n(O)(1) if membership in F can be decided in polynomial time. We solve chromatic number in O(2.2461(n)) time and domatic number in O(2.8718(n)) time. Finally, we present a family of polynomial space approximation algorithms that find a number between chi(G) and inverted right perpendicular(1 + epsilon)chi(G)inverted left perpendicular in time O(1.2209(n) + 2.2461(e-epsilon n)). (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/1477512
- author
- Björklund, Andreas LU ; Husfeldt, Thore LU and Koivisto, Mikko
- organization
- publishing date
- 2009
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- exact algorithm, set partition, inclusion-exclusion, graph coloring, zeta transform
- in
- SIAM Journal on Computing
- volume
- 39
- issue
- 2
- pages
- 546 - 563
- publisher
- Society for Industrial and Applied Mathematics
- external identifiers
-
- wos:000268859000010
- ISSN
- 0097-5397
- DOI
- 10.1137/070683933
- project
- Exact algorithms
- language
- English
- LU publication?
- yes
- id
- 0d9a4ec7-978f-44e2-af82-d188b1d99440 (old id 1477512)
- date added to LUP
- 2016-04-01 13:08:38
- date last changed
- 2021-05-05 10:34:31
@article{0d9a4ec7-978f-44e2-af82-d188b1d99440, abstract = {{Given a set N with n elements and a family F of subsets, we show how to partition N into k such subsets in 2(n) n(O)(1) time. We also consider variations of this problem where the subsets may overlap or are weighted, and we solve the decision, counting, summation, and optimization versions of these problems. Our algorithms are based on the principle of inclusion-exclusion and the zeta transform. In effect we get exact algorithms in 2(n) n(O)(1) time for several well-studied partition problems including domatic number, chromatic number, maximum k-cut, bin packing, list coloring, and the chromatic polynomial. We also have applications to Bayesian learning with decision graphs and to model-based data clustering. If only polynomial space is available, our algorithms run in time 3(n) n(O)(1) if membership in F can be decided in polynomial time. We solve chromatic number in O(2.2461(n)) time and domatic number in O(2.8718(n)) time. Finally, we present a family of polynomial space approximation algorithms that find a number between chi(G) and inverted right perpendicular(1 + epsilon)chi(G)inverted left perpendicular in time O(1.2209(n) + 2.2461(e-epsilon n)).}}, author = {{Björklund, Andreas and Husfeldt, Thore and Koivisto, Mikko}}, issn = {{0097-5397}}, keywords = {{exact algorithm; set partition; inclusion-exclusion; graph coloring; zeta transform}}, language = {{eng}}, number = {{2}}, pages = {{546--563}}, publisher = {{Society for Industrial and Applied Mathematics}}, series = {{SIAM Journal on Computing}}, title = {{Set partitioning via inclusion-exclusion}}, url = {{http://dx.doi.org/10.1137/070683933}}, doi = {{10.1137/070683933}}, volume = {{39}}, year = {{2009}}, }