### Abstract

This paper studies generalized variants of the maximum independent set problem, called the Maximum Distance-d Independent Set problem (MaxDdIS for short). For an integer d ≥ 2, a distance-d independent set of an unweighted graph G = (V,E) is a subset S ⊆ V of vertices such that for any pair of vertices u, v ∈ S, the number of edges in any path between u and v is at least d in G. Given an unweighted graph G, the goal of MaxDdIS is to find a maximum-cardinality distance-d independent set of G. In this paper, we analyze the (in)approximability of the problem on r-regular graphs (r ≥ 3) and planar graphs, as follows: (1) For every fixed integers d ≥ 3 and r ≥ 3, MaxDdIS on r-regular graphs is APX-hard. (2) We design polynomial-time O(r^{d−1})-approximation and O(r^{d−2}/d)- approximation algorithms for MaxDdIS on r-regular graphs. (3) We sharpen the above O(r^{d−2}/d)-approximation algorithms when restricted to d = r = 3, and give a polynomial-time 2-approximation algorithm for MaxD3IS on cubic graphs. (4) Finally, we show that MaxDdIS admits a polynomial-time approximation scheme (PTAS) for planar graphs.

Original language | English |
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Title of host publication | Combinatorial Optimization and Applications - 10th International Conference, COCOA 2016, Proceedings |

Editors | Minming Li, Lusheng Wang, T-H. Hubert Chan |

Publisher | Springer Verlag |

Pages | 270-284 |

Number of pages | 15 |

ISBN (Print) | 9783319487489 |

DOIs | |

Publication status | Published - Jan 1 2016 |

Event | 10th Annual International Conference on Combinatorial Optimization and Applications, COCOA 2016 - Hong Kong, China Duration: Dec 16 2016 → Dec 18 2016 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 10043 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 10th Annual International Conference on Combinatorial Optimization and Applications, COCOA 2016 |
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Country | China |

City | Hong Kong |

Period | 12/16/16 → 12/18/16 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Combinatorial Optimization and Applications - 10th International Conference, COCOA 2016, Proceedings*(pp. 270-284). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 10043 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-319-48749-6_20

**Approximability of the distance independent set problem on regular graphs and planar graphs.** / Eto, Hiroshi; Ito, Takehiro; Liu, Zhilong; Miyano, Eiji.

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

*Combinatorial Optimization and Applications - 10th International Conference, COCOA 2016, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 10043 LNCS, Springer Verlag, pp. 270-284, 10th Annual International Conference on Combinatorial Optimization and Applications, COCOA 2016, Hong Kong, China, 12/16/16. https://doi.org/10.1007/978-3-319-48749-6_20

}

TY - GEN

T1 - Approximability of the distance independent set problem on regular graphs and planar graphs

AU - Eto, Hiroshi

AU - Ito, Takehiro

AU - Liu, Zhilong

AU - Miyano, Eiji

PY - 2016/1/1

Y1 - 2016/1/1

N2 - This paper studies generalized variants of the maximum independent set problem, called the Maximum Distance-d Independent Set problem (MaxDdIS for short). For an integer d ≥ 2, a distance-d independent set of an unweighted graph G = (V,E) is a subset S ⊆ V of vertices such that for any pair of vertices u, v ∈ S, the number of edges in any path between u and v is at least d in G. Given an unweighted graph G, the goal of MaxDdIS is to find a maximum-cardinality distance-d independent set of G. In this paper, we analyze the (in)approximability of the problem on r-regular graphs (r ≥ 3) and planar graphs, as follows: (1) For every fixed integers d ≥ 3 and r ≥ 3, MaxDdIS on r-regular graphs is APX-hard. (2) We design polynomial-time O(rd−1)-approximation and O(rd−2/d)- approximation algorithms for MaxDdIS on r-regular graphs. (3) We sharpen the above O(rd−2/d)-approximation algorithms when restricted to d = r = 3, and give a polynomial-time 2-approximation algorithm for MaxD3IS on cubic graphs. (4) Finally, we show that MaxDdIS admits a polynomial-time approximation scheme (PTAS) for planar graphs.

AB - This paper studies generalized variants of the maximum independent set problem, called the Maximum Distance-d Independent Set problem (MaxDdIS for short). For an integer d ≥ 2, a distance-d independent set of an unweighted graph G = (V,E) is a subset S ⊆ V of vertices such that for any pair of vertices u, v ∈ S, the number of edges in any path between u and v is at least d in G. Given an unweighted graph G, the goal of MaxDdIS is to find a maximum-cardinality distance-d independent set of G. In this paper, we analyze the (in)approximability of the problem on r-regular graphs (r ≥ 3) and planar graphs, as follows: (1) For every fixed integers d ≥ 3 and r ≥ 3, MaxDdIS on r-regular graphs is APX-hard. (2) We design polynomial-time O(rd−1)-approximation and O(rd−2/d)- approximation algorithms for MaxDdIS on r-regular graphs. (3) We sharpen the above O(rd−2/d)-approximation algorithms when restricted to d = r = 3, and give a polynomial-time 2-approximation algorithm for MaxD3IS on cubic graphs. (4) Finally, we show that MaxDdIS admits a polynomial-time approximation scheme (PTAS) for planar graphs.

UR - http://www.scopus.com/inward/record.url?scp=85007188790&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85007188790&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-48749-6_20

DO - 10.1007/978-3-319-48749-6_20

M3 - Conference contribution

AN - SCOPUS:85007188790

SN - 9783319487489

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 270

EP - 284

BT - Combinatorial Optimization and Applications - 10th International Conference, COCOA 2016, Proceedings

A2 - Li, Minming

A2 - Wang, Lusheng

A2 - Chan, T-H. Hubert

PB - Springer Verlag

ER -