### Abstract

Let HDd(p, q) denote the minimal size of a transversal that can always be guaranteed for a family of compact convex sets in Rd which satisfy the (p, q)-property (p ≥ q ≥ d + 1). In a celebrated proof of the Hadwiger-Debrunner conjecture, Alon and Kleitman proved that HDd(p, q) exists for all p ≥ q ≥ d + 1. Specifically, they prove that HDd(p, d + 1) is O(p^{d2+d}). This paper has two parts. In the first part we present several improved bounds on HDd(p, q). In particular, we obtain the first near tight estimate of HDd(p, q) for an extended range of values of (p, q) since the 1957 Hadwiger-Debrunner theorem. In the second part we prove a (p, 2)-theorem for families in R2 with union complexity below a specific quadratic bound. Based on this, we introduce a polynomial time constant factor approximation algorithm for MAX-CLIQUE of intersection graphs of convex sets satisfying this property. It is not likely that our constant factor approximation can be improved to a PTAS as MAX-CLIQUE for intersection graphs of fat ellipses is known to be APX-HARD and fat ellipses have sub-quadratic union complexity.

Original language | English |
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Title of host publication | 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017 |

Publisher | Association for Computing Machinery |

Pages | 2254-2263 |

Number of pages | 10 |

ISBN (Electronic) | 9781611974782 |

Publication status | Published - Jan 1 2017 |

Event | 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017 - Barcelona, Spain Duration: Jan 16 2017 → Jan 19 2017 |

### Other

Other | 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017 |
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Country | Spain |

City | Barcelona |

Period | 1/16/17 → 1/19/17 |

### Fingerprint

### ASJC Scopus subject areas

- Software
- Mathematics(all)

### Cite this

*28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017*(pp. 2254-2263). Association for Computing Machinery.

**On max-clique for intersection graphs of sets and the hadwiger-debrunner numbers.** / Keller, Chaya; Smorodinsky, Shakhar; Tardos, G.

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

*28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017.*Association for Computing Machinery, pp. 2254-2263, 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, 1/16/17.

}

TY - GEN

T1 - On max-clique for intersection graphs of sets and the hadwiger-debrunner numbers

AU - Keller, Chaya

AU - Smorodinsky, Shakhar

AU - Tardos, G.

PY - 2017/1/1

Y1 - 2017/1/1

N2 - Let HDd(p, q) denote the minimal size of a transversal that can always be guaranteed for a family of compact convex sets in Rd which satisfy the (p, q)-property (p ≥ q ≥ d + 1). In a celebrated proof of the Hadwiger-Debrunner conjecture, Alon and Kleitman proved that HDd(p, q) exists for all p ≥ q ≥ d + 1. Specifically, they prove that HDd(p, d + 1) is O(pd2+d). This paper has two parts. In the first part we present several improved bounds on HDd(p, q). In particular, we obtain the first near tight estimate of HDd(p, q) for an extended range of values of (p, q) since the 1957 Hadwiger-Debrunner theorem. In the second part we prove a (p, 2)-theorem for families in R2 with union complexity below a specific quadratic bound. Based on this, we introduce a polynomial time constant factor approximation algorithm for MAX-CLIQUE of intersection graphs of convex sets satisfying this property. It is not likely that our constant factor approximation can be improved to a PTAS as MAX-CLIQUE for intersection graphs of fat ellipses is known to be APX-HARD and fat ellipses have sub-quadratic union complexity.

AB - Let HDd(p, q) denote the minimal size of a transversal that can always be guaranteed for a family of compact convex sets in Rd which satisfy the (p, q)-property (p ≥ q ≥ d + 1). In a celebrated proof of the Hadwiger-Debrunner conjecture, Alon and Kleitman proved that HDd(p, q) exists for all p ≥ q ≥ d + 1. Specifically, they prove that HDd(p, d + 1) is O(pd2+d). This paper has two parts. In the first part we present several improved bounds on HDd(p, q). In particular, we obtain the first near tight estimate of HDd(p, q) for an extended range of values of (p, q) since the 1957 Hadwiger-Debrunner theorem. In the second part we prove a (p, 2)-theorem for families in R2 with union complexity below a specific quadratic bound. Based on this, we introduce a polynomial time constant factor approximation algorithm for MAX-CLIQUE of intersection graphs of convex sets satisfying this property. It is not likely that our constant factor approximation can be improved to a PTAS as MAX-CLIQUE for intersection graphs of fat ellipses is known to be APX-HARD and fat ellipses have sub-quadratic union complexity.

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M3 - Conference contribution

AN - SCOPUS:85016177978

SP - 2254

EP - 2263

BT - 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017

PB - Association for Computing Machinery

ER -