On the Carathéodory number of interval and graph convexities

Authors:
Mitre C. Dourado;Dieter Rautenbach;Vinícius Fernandes Dos Santos;Philipp M. Schäfer;Jayme L. Szwarcfiter
Affiliations:
Instituto de Matemática, NCE, and COPPE, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, Brazil;Institut für Optimierung und Operations Research, Universität Ulm, Ulm, Germany;Instituto de Matemática, NCE, and COPPE, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, Brazil;Institut für Optimierung und Operations Research, Universität Ulm, Ulm, Germany;Instituto de Matemática, NCE, and COPPE, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, Brazil
Venue:
Theoretical Computer Science
Year:
2013

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Abstract

Inspired by a result of Caratheodory [Uber den Variabilitatsbereich der Fourierschen Konstanten von positiven harmonischen Funktionen, Rend. Circ. Mat. Palermo 32 (1911) 193-217], the Caratheodory number of a convexity space is defined as the smallest integer k such that for every subset U of the ground set V and every element u in the convex hull of U, there is a subset F of U with at most k elements such that u in the convex hull of F. We study the Caratheodory number for generalized interval convexities and for convexity spaces derived from finite graphs. We establish structural properties, bounds, and hardness results.