### Abstract

It is an old open question in algorithmic graph theory 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., SODA 2014, pp. 570-581, 2014]. Here we study the existence of subexponential-time algorithms for the problem: by generalizing an earlier result of Randerath and Schiermeyer for t = 5 [Discrete Appl. Math., 158 (2010), pp. 1041-1044], we show that for any t ≥ 5, there is an algorithm for Maximum Independent Set on P_{t}-free graphs whose running time is subexponential in the number of vertices. 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 number of edges): If every component of H is a path, then d-Scattered Set on H-free graphs with n vertices and m edges can be solved in time 2^{(n+m)} ^{1-O(1/|V (H)|)}, even if d is part of the input. Otherwise, assuming ETH, there is no 2^{o(n+m)}-time algorithm for d-Scattered Set for any fixed d ≥ 3 on H-free graphs with n-vertices and m-edges.

Original language | English |
---|---|

Title of host publication | 11th International Symposium on Parameterized and Exact Computation, IPEC 2016 |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

Volume | 63 |

ISBN (Electronic) | 9783959770231 |

DOIs | |

Publication status | Published - Feb 1 2017 |

Event | 11th International Symposium on Parameterized and Exact Computation, IPEC 2016 - Aarhus, Denmark Duration: Aug 24 2016 → Aug 26 2016 |

### Other

Other | 11th International Symposium on Parameterized and Exact Computation, IPEC 2016 |
---|---|

Country | Denmark |

City | Aarhus |

Period | 8/24/16 → 8/26/16 |

### Fingerprint

### Keywords

- H-free graphs
- Independent set
- Scattered set
- Subexponential algorithms

### ASJC Scopus subject areas

- Software

### Cite this

*11th International Symposium on Parameterized and Exact Computation, IPEC 2016*(Vol. 63). [3] Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.IPEC.2016.3

**H-free graphs, Independent Sets, and subexponential-time algorithms.** / Bacsó, Gábor; Marx, Dániel; Tuza, Z.

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

*11th International Symposium on Parameterized and Exact Computation, IPEC 2016.*vol. 63, 3, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 11th International Symposium on Parameterized and Exact Computation, IPEC 2016, Aarhus, Denmark, 8/24/16. https://doi.org/10.4230/LIPIcs.IPEC.2016.3

}

TY - GEN

T1 - H-free graphs, Independent Sets, and subexponential-time algorithms

AU - Bacsó, Gábor

AU - Marx, Dániel

AU - Tuza, Z.

PY - 2017/2/1

Y1 - 2017/2/1

N2 - It is an old open question in algorithmic graph theory to determine the complexity of the Maximum Independent Set problem on Pt-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., SODA 2014, pp. 570-581, 2014]. Here we study the existence of subexponential-time algorithms for the problem: by generalizing an earlier result of Randerath and Schiermeyer for t = 5 [Discrete Appl. Math., 158 (2010), pp. 1041-1044], we show that for any t ≥ 5, there is an algorithm for Maximum Independent Set on Pt-free graphs whose running time is subexponential in the number of vertices. 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 number of edges): If every component of H is a path, then d-Scattered Set on H-free graphs with n vertices and m edges can be solved in time 2(n+m) 1-O(1/|V (H)|), even if d is part of the input. Otherwise, assuming ETH, there is no 2o(n+m)-time algorithm for d-Scattered Set for any fixed d ≥ 3 on H-free graphs with n-vertices and m-edges.

AB - It is an old open question in algorithmic graph theory to determine the complexity of the Maximum Independent Set problem on Pt-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., SODA 2014, pp. 570-581, 2014]. Here we study the existence of subexponential-time algorithms for the problem: by generalizing an earlier result of Randerath and Schiermeyer for t = 5 [Discrete Appl. Math., 158 (2010), pp. 1041-1044], we show that for any t ≥ 5, there is an algorithm for Maximum Independent Set on Pt-free graphs whose running time is subexponential in the number of vertices. 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 number of edges): If every component of H is a path, then d-Scattered Set on H-free graphs with n vertices and m edges can be solved in time 2(n+m) 1-O(1/|V (H)|), even if d is part of the input. Otherwise, assuming ETH, there is no 2o(n+m)-time algorithm for d-Scattered Set for any fixed d ≥ 3 on H-free graphs with n-vertices and m-edges.

KW - H-free graphs

KW - Independent set

KW - Scattered set

KW - Subexponential algorithms

UR - http://www.scopus.com/inward/record.url?scp=85014696217&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85014696217&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.IPEC.2016.3

DO - 10.4230/LIPIcs.IPEC.2016.3

M3 - Conference contribution

AN - SCOPUS:85014696217

VL - 63

BT - 11th International Symposium on Parameterized and Exact Computation, IPEC 2016

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

ER -