# Subexponential-Time Algorithms for Maximum Independent Set in Pt-Free and Broom-Free Graphs

Bacsó, Gábor and Lokshtanov, D and Marx, Dániel and Pilipczuk, M and Tuza, Zsolt
(2019)
*Subexponential-Time Algorithms for Maximum Independent Set in Pt-Free and Broom-Free Graphs.*
ALGORITHMICA, 81 (2).
pp. 421-438.
ISSN 0178-4617
10.1007/s00453-018-0479-5

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## Abstract

In algorithmic graph theory, a classic open question is to determine the complexity of the Maximum Independent Set problem on P-t-free graphs, that is, on graphs not containing any induced path on t vertices. So far, polynomial-time algorithms are known only for t <= 5 (Lokshtanov et al., in: Proceedings of the twenty-fifth annual ACM-SIAM symposium on discrete algorithms, SODA 2014, Portland, OR, USA, January 5-7, 2014, pp 570-581, 2014), and an algorithm for t = 6 announced recently (Grzesik et al. in Polynomial-time algorithm for maximum weight independent set on P-6-free graphs. CoRR, arXiv:1707.05491, 2017). Here we study the existence of subexponential-time algorithms for the problem: we show that for any t >= 1, there is an algorithm for Maximum Independent Set on P-t-free graphs whose running time is subexponential in the number of vertices. Even for the weighted version MWIS, the problem is solvable in 2(O(root tn log n)) time on P-t-free graphs. For approximation of MIS in broom-free graphs, a similar time bound is proved. Scattered Set is the generalization of Maximum Independent Set where the vertices of the solution are required to be at distance at least d from each other. We give a complete characterization of those graphs H for which d-Scattered Set on H-free graphs can be solved in time subexponential in the size of the input (that is, in the number of vertices plus the number of edges):

Item Type: | Article |
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Subjects: | Q Science > QA Mathematics and Computer Science > QA75 Electronic computers. Computer science / számítástechnika, számítógéptudomány |

Divisions: | Informatics Laboratory |

SWORD Depositor: | MTMT Injector |

Depositing User: | MTMT Injector |

Date Deposited: | 14 Oct 2019 09:08 |

Last Modified: | 14 Oct 2019 09:08 |

URI: | http://eprints.sztaki.hu/id/eprint/9823 |

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