Author:
Castro-Silva Davi,de Oliveira Filho Fernando Mário,Slot Lucas,Vallentin Frank
Abstract
AbstractThe theta body of a graph, introduced by Grötschel, Lovász, and Schrijver (in 1986), is a tractable relaxation of the independent-set polytope derived from the Lovász theta number. In this paper, we recursively extend the theta body, and hence the theta number, to hypergraphs. We obtain fundamental properties of this extension and relate it to the high-dimensional Hoffman bound of Filmus, Golubev, and Lifshitz. We discuss two applications: triangle-free graphs and Mantel’s theorem, and bounds on the density of triangle-avoiding sets in the Hamming cube.
Publisher
Springer Science and Business Media LLC
Subject
Computational Mathematics,Discrete Mathematics and Combinatorics
Reference25 articles.
1. Bachoc, C., DeCorte, P.E.B., de Oliveira Filho, F.M., Vallentin, F.: Spectral bounds for the independence ratio and the chromatic number of an operator. Israel J. Math. 202, 227–254 (2014)
2. Bachoc, C., Gijswijt, D.C., Schrijver, A., Vallentin, F.: Invariant semidefinite programs. In: Anjos, M.F., Lasserre, J.B. (eds.) Handbook on semidefinite, conic, and polynomial optimization, pp. 219–269. Springer, New York (2012)
3. Castro-Silva, D., de Oliveira Filho, F.M., Slot, L., Vallentin, F.: A recursive Lovász theta number for simplex-avoiding sets. Proceedings of the American Mathematical Society 150, 3307–3322 (2022)
4. de Klerk, E., Vallentin, F.: On the turing model complexity of interior point methods for semidefinite programming. SIAM J. Optim. 26, 1944–1961 (2016)
5. de Oliveira Filho, F.M., Vallentin, F.: Computing upper bounds for the packing density of congruent copies of a convex body. In: Ambrus, G., Bárány, I., Böröczky, K.J., Fejestóth, G., Pach, J. (eds.) New Trends in Intuitive Geometry Bolyai Society Mathematical Studies, pp. 155–188. Springer, Berlin (2019)