Parameterized algorithms for graph partitioning problems

Hadas Shachnai; Meirav Zehavi

doi:10.1007/978-3-319-12340-0_32

Parameterized algorithms for graph partitioning problems

Hadas Shachnai, Meirav Zehavi

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

6 Scopus citations

Abstract

We study a broad class of graph partitioning problems, where each problem is specified by a graph G = (V,E), and parameters k and p. We seek a subset U ⊆ V of size k, such that α₁m₁ + α₂m₂ is at most (or at least) p, where α₁, α₂ ∈ R are constants defining the problem, and m₁,m₂ are the cardinalities of the edge sets having both endpoints, and exactly one endpoint, in U, respectively. This class of fixed-cardinality graph partitioning problems (FGPPs) encompasses Max (k, n − k)-Cut, Min k-Vertex Cover, k-Densest Subgraph, and k- Sparsest Subgraph.

Our main result is an O∗ (4^k+o(k)Δk) algorithm for any problem in this class, where Δ ≥ 1 is the maximum degree in the input graph. This resolves an open question posed by Bonnet et al. [IPEC 2013]. We obtain faster algorithms for certain subclasses of FGPPs, parameterized by p, or by (k + p). In particular, we give an O∗(4^p+o(p)) time algorithm for Max (k, n−k)-Cut, thus improving significantly the best known O∗(p^p) time algorithm.

Original language	English
Title of host publication	Graph-Theoretic Concepts in Computer Science - 40th International Workshop, WG 2014, Revised Selected Papers
Editors	Dieter Kratsch, Ioan Todinca
Publisher	Springer Verlag
Pages	384-395
Number of pages	12
ISBN (Electronic)	9783319123394
DOIs	https://doi.org/10.1007/978-3-319-12340-0_32
State	Published - 1 Jan 2014
Externally published	Yes
Event	40th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2014 - Orléans, France Duration: 25 Jun 2014 → 27 Jun 2014

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	8747
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	40th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2014
Country/Territory	France
City	Orléans
Period	25/06/14 → 27/06/14

ASJC Scopus subject areas

Theoretical Computer Science
Computer Science (all)

Access to Document

10.1007/978-3-319-12340-0_32

Cite this

Shachnai, H., & Zehavi, M. (2014). Parameterized algorithms for graph partitioning problems. In D. Kratsch, & I. Todinca (Eds.), Graph-Theoretic Concepts in Computer Science - 40th International Workshop, WG 2014, Revised Selected Papers (pp. 384-395). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8747). Springer Verlag. https://doi.org/10.1007/978-3-319-12340-0_32

Shachnai, Hadas ; Zehavi, Meirav. / Parameterized algorithms for graph partitioning problems. Graph-Theoretic Concepts in Computer Science - 40th International Workshop, WG 2014, Revised Selected Papers. editor / Dieter Kratsch ; Ioan Todinca. Springer Verlag, 2014. pp. 384-395 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

@inproceedings{dd1288a3b3c24d4392e70ef37d415851,

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abstract = "We study a broad class of graph partitioning problems, where each problem is specified by a graph G = (V,E), and parameters k and p. We seek a subset U ⊆ V of size k, such that α1m1 + α2m2 is at most (or at least) p, where α1, α2 ∈ R are constants defining the problem, and m1,m2 are the cardinalities of the edge sets having both endpoints, and exactly one endpoint, in U, respectively. This class of fixed-cardinality graph partitioning problems (FGPPs) encompasses Max (k, n − k)-Cut, Min k-Vertex Cover, k-Densest Subgraph, and k- Sparsest Subgraph.Our main result is an O∗ (4k+o(k)Δk) algorithm for any problem in this class, where Δ ≥ 1 is the maximum degree in the input graph. This resolves an open question posed by Bonnet et al. [IPEC 2013]. We obtain faster algorithms for certain subclasses of FGPPs, parameterized by p, or by (k + p). In particular, we give an O∗(4p+o(p)) time algorithm for Max (k, n−k)-Cut, thus improving significantly the best known O∗(pp) time algorithm.",

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Shachnai, H & Zehavi, M 2014, Parameterized algorithms for graph partitioning problems. in D Kratsch & I Todinca (eds), Graph-Theoretic Concepts in Computer Science - 40th International Workshop, WG 2014, Revised Selected Papers. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 8747, Springer Verlag, pp. 384-395, 40th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2014, Orléans, France, 25/06/14. https://doi.org/10.1007/978-3-319-12340-0_32

Parameterized algorithms for graph partitioning problems. / Shachnai, Hadas; Zehavi, Meirav.
Graph-Theoretic Concepts in Computer Science - 40th International Workshop, WG 2014, Revised Selected Papers. ed. / Dieter Kratsch; Ioan Todinca. Springer Verlag, 2014. p. 384-395 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8747).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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N2 - We study a broad class of graph partitioning problems, where each problem is specified by a graph G = (V,E), and parameters k and p. We seek a subset U ⊆ V of size k, such that α1m1 + α2m2 is at most (or at least) p, where α1, α2 ∈ R are constants defining the problem, and m1,m2 are the cardinalities of the edge sets having both endpoints, and exactly one endpoint, in U, respectively. This class of fixed-cardinality graph partitioning problems (FGPPs) encompasses Max (k, n − k)-Cut, Min k-Vertex Cover, k-Densest Subgraph, and k- Sparsest Subgraph.Our main result is an O∗ (4k+o(k)Δk) algorithm for any problem in this class, where Δ ≥ 1 is the maximum degree in the input graph. This resolves an open question posed by Bonnet et al. [IPEC 2013]. We obtain faster algorithms for certain subclasses of FGPPs, parameterized by p, or by (k + p). In particular, we give an O∗(4p+o(p)) time algorithm for Max (k, n−k)-Cut, thus improving significantly the best known O∗(pp) time algorithm.

AB - We study a broad class of graph partitioning problems, where each problem is specified by a graph G = (V,E), and parameters k and p. We seek a subset U ⊆ V of size k, such that α1m1 + α2m2 is at most (or at least) p, where α1, α2 ∈ R are constants defining the problem, and m1,m2 are the cardinalities of the edge sets having both endpoints, and exactly one endpoint, in U, respectively. This class of fixed-cardinality graph partitioning problems (FGPPs) encompasses Max (k, n − k)-Cut, Min k-Vertex Cover, k-Densest Subgraph, and k- Sparsest Subgraph.Our main result is an O∗ (4k+o(k)Δk) algorithm for any problem in this class, where Δ ≥ 1 is the maximum degree in the input graph. This resolves an open question posed by Bonnet et al. [IPEC 2013]. We obtain faster algorithms for certain subclasses of FGPPs, parameterized by p, or by (k + p). In particular, we give an O∗(4p+o(p)) time algorithm for Max (k, n−k)-Cut, thus improving significantly the best known O∗(pp) time algorithm.

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Shachnai H, Zehavi M. Parameterized algorithms for graph partitioning problems. In Kratsch D, Todinca I, editors, Graph-Theoretic Concepts in Computer Science - 40th International Workshop, WG 2014, Revised Selected Papers. Springer Verlag. 2014. p. 384-395. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/978-3-319-12340-0_32

Parameterized algorithms for graph partitioning problems

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