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Register a new userMatchings, hypergraphs, association schemes, and semidefinite optimization
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Added by
Yu Hin (Gary) Au
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We utilize association schemes to analyze the quality of semidefinite programming (SDP) based convex relaxations of integral packing and covering polyhedra determined by matchings in hypergraphs. As a by-product of our approach, we obtain bounds on the clique and stability numbers of some regular graphs reminiscent of classical bounds by Delsarte and Hoffman. We determine exactly or provide bounds on the performance of Lov{a}sz-Schrijver SDP hierarchy, and illustrate the usefulness of obtaining commutative subschemes from non-commutative schemes via contraction in this context.
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A graph $G$ whose edges are coloured (not necessarily properly) contains a full rainbow matching if there is a matching $M$ that contains exactly one edge of each colour. We refute several conjectures on matchings in hypergraphs and full rainbow matchings in graphs, made by Aharoni and Berger and others.
An association scheme is called quasi-thin if the valency of each its basic relation is one or two. A quasi-thin scheme is Kleinian if the thin residue of it forms a Klein group with respect to the relation product. It is proved that any Kleinian scheme arises from near-pencil on~$3$ points, or affine or projective plane of order~$2$. The main result is that any non-Kleinian quasi-thin scheme a) is the two-orbit scheme of a suitable permutation group, and b) is characterized up to isomorphism by its intersection number array. An infinite family of Kleinian quasi-thin schemes for which neither a) nor b) holds is also constructed.
We construct twelve infinite families of pseudocyclic and non-amorphic association schemes, in which each nontrivial relation is a strongly regular graph. Three of the twelve families generalize the counterexamples to A. V. Ivanovs conjecture by Ikuta and Munemasa [15].
The main goal of the paper is to establish a sufficient condition for a two-valenced association scheme to be schurian and separable. To this end, an analog of the Desargues theorem is introduced for a noncommutative geometry defined by the scheme in question. It turns out that if the geometry has enough many Desarguesian configurations, then under a technical condition the scheme is schurian and separable. This result enables us to give short proofs for known statements on the schurity and separability of quasi-thin and pseudocyclic schemes. Moreover, by the same technique we prove a new result: given a prime $p$, any ${1,p}$-scheme with thin residue isomorphic to an elementary abelian $p$-group of rank greater than two, is schurian and separable.
{Let ${Cal X}$ be a self-dual P-polynomial association scheme. Then there are at most 12 diagonal matrices $T$ such that $(PT)^3=I$. Moreover, all of the solutions for the classical infinite families of such schemes (including the Hamming scheme) are classified.
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