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

Chaya Keller, Shakhar Smorodinsky, G. Tardos

Research output: Chapter in Book/Report/Conference proceedingConference contribution

10 Citations (Scopus)

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(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.

Original languageEnglish
Title of host publication28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017
PublisherAssociation for Computing Machinery
Pages2254-2263
Number of pages10
ISBN (Electronic)9781611974782
Publication statusPublished - Jan 1 2017
Event28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017 - Barcelona, Spain
Duration: Jan 16 2017Jan 19 2017

Other

Other28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017
CountrySpain
CityBarcelona
Period1/16/171/19/17

Fingerprint

Intersection Graphs
Oils and fats
Clique
Union
Hadwiger's Conjecture
Compact Convex Set
Approximation algorithms
Time Constant
Theorem
Convex Sets
Approximation Algorithms
Polynomial time
Likely
Polynomials
Denote
Approximation
Estimate
Range of data
Family

ASJC Scopus subject areas

  • Software
  • Mathematics(all)

Cite this

Keller, C., Smorodinsky, S., & Tardos, G. (2017). On max-clique for intersection graphs of sets and the hadwiger-debrunner numbers. In 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.

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

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Keller, C, Smorodinsky, S & Tardos, G 2017, On max-clique for intersection graphs of sets and the hadwiger-debrunner numbers. in 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.
Keller C, Smorodinsky S, Tardos G. On max-clique for intersection graphs of sets and the hadwiger-debrunner numbers. In 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017. Association for Computing Machinery. 2017. p. 2254-2263
Keller, Chaya ; Smorodinsky, Shakhar ; Tardos, G. / On max-clique for intersection graphs of sets and the hadwiger-debrunner numbers. 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017. Association for Computing Machinery, 2017. pp. 2254-2263
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