On Semidefinite Programming Relaxations of Association Schemes With Application to Combinatorial Optimization Problems

AMS classification: 90C22, 20Cxx, 70-08traveling salesman problem;maximum bisection;semidefinite programming;association schemes

Solving SDP's in Non-commutative Algebras Part I: The Dual-Scaling Algorithm

Semidefinite programming (SDP) may be viewed as an extension of linear programming (LP), and most interior point methods (IPM s) for LP can be extended to solve SDP problems.However, it is far more difficult to exploit data structures (especially sparsity) in the SDP case.In this paper we will look at the data structure where the SDP data matrices lie in a low dimensional matrix algebra.This data structure occurs in several applications, including the lower bounding of the stability number in certain graphs and the crossing number in complete bipartite graphs.We will show that one can reduce the linear algebra involved in an iteration of an IPM to involve matrices of the size of the dimension of the matrix algebra only.In other words, the original sizes of the data matrices do not appear in the computational complexity bound.In particular, we will work out the details for the dual scaling algorithm, since a dual method is most suitable for the types of applications we have in mind.semidefinite programming;matrix algebras;dual scaling algorithm;exploiting data structure

A note on the stability number of an orthogonality graph

We consider the orthogonality graph Omega(n) with 2^n vertices corresponding to the 0-1 n-vectors, two vertices adjacent if and only if the Hamming distance between them is n/2. We show that the stability number of Omega(16) is alpha(Omega(16))= 2304, thus proving a conjecture by Galliard. The main tool we employ is a recent semidefinite programming relaxation for minimal distance binary codes due to Schrijver. As well, we give a general condition for Delsarte bound on the (co)cliques in graphs of relations of association schemes to coincide with the ratio bound, and use it to show that for Omega(n) the latter two bounds are equal to 2^n/n.Comment: 10 pages, LaTeX, 1 figure, companion Matlab code. Misc. misprints fixed and references update

A Note on the Stability Number of an Orthogonality Graph

We consider the orthogonality graph (n) with 2n vertices corresponding to the vectors {0, 1}n, two vertices adjacent if and only if the Hamming distance between them is n/2.We show that, for n = 16, the stability number of (n) is ( (16)) = 2304, thus proving a conjecture by Galliard [7].The main tool we employ is a recent semidefinite programming relaxation for minimal distance binary codes due to Schrijver [16].Moreover, we give a general condition for Delsarte bound on the (co)cliques in graphs of relations of association schemes to coincide with the ratio bound, and use it to show that for (n) the latter two bounds are equal to 2n/n.C0;C61

On Semidefinite Programming Relaxations of the Travelling Salesman Problem (Replaced by DP 2008-96)

AMS classification: 90C22, 20Cxx, 70-08traveling salesman problem;semidefinite programming;quadratic as- signment problem

Phenomenology of an extended IDM with loop-generated fermion mass hierarchies

We perform a comprehensive analysis of the most distinctive and important phenomenological implications of the recently proposed mechanism of sequential loop generation of strong hierarchies in the Standard Model (SM) fermion mass spectra. This mechanism is consistently realized at the level of renormalizable interactions in an extended variant of the Inert Higgs Doublet model, possessing the additional $Z_{2}^{(1)}\times Z_{2}^{(2)}$ discrete and $U_{1X}$ gauge family symmetries, while the matter sectors of the SM are extended by means of $SU_{2L}$-singlet scalars, heavy vector-like leptons and quarks, as well as right-handed neutrinos. We thoroughly analyze the most stringent constraints on the model parameter space, coming from the $Z^{\prime }$ collider searches, related to the anomaly in lepton universality, and the muon anomalous magnetic moment, as well as provide benchmark points for further tests of the model and discuss possible "standard candle" signatures relevant for future explorations.Comment: Version accepted for publication in EPJC. arXiv admin note: text overlap with arXiv:1901.0276