The Independence Number of the Orthogonality Graph in Dimension 2k2^k

We determine the independence number of the orthogonality graph on $2^k$-dimensional hypercubes. This answers a question by Galliard from 2001 which is motivated by a problem in quantum information theory. Our method is a modification of a rank argument due to Frankl who showed the analogous result for $4p^k$-dimensional hypercubes, where $p$ is an odd prime.Comment: 3 pages, accepted by Combinatorica, fixed a minor typo spotted by Peter Si

Block-diagonal semidefinite programming hierarchies for 0/1 programming

Lovasz and Schrijver, and later Lasserre, proposed hierarchies of semidefinite programming relaxations for general 0/1 linear programming problems. In this paper these two constructions are revisited and two new, block-diagonal hierarchies are proposed. They have the advantage of being computationally less costly while being at least as strong as the Lovasz-Schrijver hierarchy. Our construction is applied to the stable set problem and experimental results for Paley graphs are reported.Comment: 11 pages, (v2) revision based on suggestions by referee, computation of N+(TH(P_q)) included in Table

A semidefinite programming hierarchy for packing problems in discrete geometry

Packing problems in discrete geometry can be modeled as finding independent sets in infinite graphs where one is interested in independent sets which are as large as possible. For finite graphs one popular way to compute upper bounds for the maximal size of an independent set is to use Lasserre's semidefinite programming hierarchy. We generalize this approach to infinite graphs. For this we introduce topological packing graphs as an abstraction for infinite graphs coming from packing problems in discrete geometry. We show that our hierarchy converges to the independence number.Comment: (v2) 25 pages, revision based on suggestions by referee, accepted in Mathematical Programming Series B special issue on polynomial optimizatio

Exact Completeness of LP Hierarchies for Linear Codes

Determining the maximum size $A_2(n,d)$ of a binary code of blocklength $n$ and distance $d$ remains an elusive open question even when restricted to the important class of linear codes. Recently, two linear programming hierarchies extending Delsarte's LP were independently proposed to upper bound $A_2^{\text{Lin}}(n,d)$ (the analogue of $A_2(n,d)$ for linear codes). One of these hierarchies, by the authors, was shown to be approximately complete in the sense that the hierarchy converges to $A_2^{\text{Lin}}(n,d)$ as the level grows beyond $n^2$. Despite some structural similarities, not even approximate completeness was known for the other hierarchy by Loyfer and Linial. In this work, we prove that both hierarchies recover the exact value of $A_2^{\text{Lin}}(n,d)$ at level $n$. We also prove that at this level the polytope of Loyfer and Linial is integral.Even though these hierarchies seem less powerful than general hierarchies such as Sum-of-Squares, we show that they have enough structure to yield exact completeness via pseudoprobabilities.Comment: 19 page

Lecture notes: Semidefinite programs and harmonic analysis

Lecture notes for the tutorial at the workshop HPOPT 2008 - 10th International Workshop on High Performance Optimization Techniques (Algebraic Structure in Semidefinite Programming), June 11th to 13th, 2008, Tilburg University, The Netherlands.Comment: 31 page