InforLorV4, Main, Exploration, bibRecord, 000201

Minimum Degree up to Local Complementation: Bounds, Parameterized Complexity, and Exact Algorithms

Identifieur interne : 000201 ( Main/Exploration ); précédent : 000200; suivant : 000202

Minimum Degree up to Local Complementation: Bounds, Parameterized Complexity, and Exact Algorithms

Auteurs : David Cattanéo [France] ; Simon Perdrix [France]

Source :

RBID : Hal:hal-01132843

Abstract

The local minimum degree of a graph is the minimum degree that can be reached by means of local complementation. For any n, there exist graphs of order n which have a local minimum degree at least 0.189n, or at least 0.110n when restricted to bipartite graphs. Regarding the upper bound, we show that for any graph of order n, its local minimum degree is at most 3n/8+o(n) and n/4+o(n) for bipartite graphs, improving the known n/2 upper bound. We also prove that the local minimum degree is smaller than half of the vertex cover number (up to a logarithmic term). The local minimum degree problem is NP-Complete and hard to approximate. We show that this problem, even when restricted to bipartite graphs, is in W[2] and FPT-equivalent to the EvenSet problem, which W[1]-hardness is a long standing open question. Finally, we show that the local minimum degree is computed by a O*(1.938^n)-algorithm, and a O*(1.466^n)-algorithm for the bipartite graphs.

Url:

https://hal.archives-ouvertes.fr/hal-01132843

DOI: 10.1007/978-3-662-48971-0_23

Affiliations:

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to stream Hal, to step Corpus: 003196
to stream Hal, to step Curation: 003196
to stream Hal, to step Checkpoint: 000172
to stream Main, to step Merge: 000201
to stream Main, to step Curation: 000201

Le document en format XML

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<front><div type="abstract" xml:lang="en">The local minimum degree of a graph is the minimum degree that can be reached by means of local complementation. For any n, there exist graphs of order n which have a local minimum degree at least 0.189n, or at least 0.110n when restricted to bipartite graphs. Regarding the upper bound, we show that for any graph of order n, its local minimum degree is at most 3n/8+o(n) and n/4+o(n) for bipartite graphs, improving the known n/2 upper bound. We also prove that the local minimum degree is smaller than half of the vertex cover number (up to a logarithmic term). The local minimum degree problem is NP-Complete and hard to approximate. We show that this problem, even when restricted to bipartite graphs, is in W[2] and FPT-equivalent to the EvenSet problem, which W[1]-hardness is a long standing open question. Finally, we show that the local minimum degree is computed by a O*(1.938^n)-algorithm, and a O*(1.466^n)-algorithm for the bipartite graphs.</div>
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Minimum Degree up to Local Complementation: Bounds, Parameterized Complexity, and Exact Algorithms

Minimum Degree up to Local Complementation: Bounds, Parameterized Complexity, and Exact Algorithms

Source :

Abstract

Links toward previous steps (curation, corpus...)

Le document en format XML

Pour manipuler ce document sous Unix (Dilib)

Pour mettre un lien sur cette page dans le réseau Wicri