46 research outputs found
A new graph parameter related to bounded rank positive semidefinite matrix completions
The Gram dimension \gd(G) of a graph is the smallest integer
such that any partial real symmetric matrix, whose entries are specified on the
diagonal and at the off-diagonal positions corresponding to edges of , can
be completed to a positive semidefinite matrix of rank at most (assuming a
positive semidefinite completion exists). For any fixed the class of graphs
satisfying \gd(G) \le k is minor closed, hence it can characterized by a
finite list of forbidden minors. We show that the only minimal forbidden minor
is for and that there are two minimal forbidden minors:
and for . We also show some close connections to
Euclidean realizations of graphs and to the graph parameter of
\cite{H03}. In particular, our characterization of the graphs with \gd(G)\le
4 implies the forbidden minor characterization of the 3-realizable graphs of
Belk and Connelly \cite{Belk,BC} and of the graphs with of van
der Holst \cite{H03}.Comment: 31 pages, 6 Figures. arXiv admin note: substantial text overlap with
arXiv:1112.596
Forbidden minor characterizations for low-rank optimal solutions to semidefinite programs over the elliptope
We study a new geometric graph parameter \egd(G), defined as the smallest
integer for which any partial symmetric matrix which is completable to
a correlation matrix and whose entries are specified at the positions of the
edges of , can be completed to a matrix in the convex hull of correlation
matrices of \rank at most . This graph parameter is motivated by its
relevance to the problem of finding low rank solutions to semidefinite programs
over the elliptope, and also by its relevance to the bounded rank Grothendieck
constant. Indeed, \egd(G)\le r if and only if the rank- Grothendieck
constant of is equal to 1. We show that the parameter \egd(G) is minor
monotone, we identify several classes of forbidden minors for \egd(G)\le r
and we give the full characterization for the case . We also show an upper
bound for \egd(G) in terms of a new tree-width-like parameter \sla(G),
defined as the smallest for which is a minor of the strong product of a
tree and . We show that, for any 2-connected graph on at
least 6 nodes, \egd(G)\le 2 if and only if \sla(G)\le 2.Comment: 33 pages, 8 Figures. In its second version, the paper has been
modified to accommodate the suggestions of the referees. Furthermore, the
title has been changed since we feel that the new title reflects more
accurately the content and the main results of the pape
Positive semidefinite matrix completion, universal rigidity and the Strong Arnold Property
This paper addresses the following three topics: positive semidefinite (psd) matrix completions, universal rigidity of frameworks, and the Strong Arnold Property (SAP). We show some strong connections among these topics, using semidefinite programming as unifying theme. Our main contribution is a sufficient condition for constructing partial psd matrices which admit a unique completion to a full psd matrix. Such partial matrices are an essential tool in the study of the Gram dimension \gd(G) of a graph G, a recently studied graph parameter related to the low psd matrix completion problem. Additionally, we derive an elementary proof of Connelly's sufficient condition for universal rigidity of tensegrity frameworks and we investigate the links between these two sufficient conditions. We also give a geometric characterization of psd matrices satisfying the Strong Arnold Property in terms of nondegeneracy of an associated semidefinite program, which we use to establish some links between the Gram dimension \gd(\cdot) and the Colin de Verdi\`ere type graph parameter ν=(⋅)
Positive semidefinite matrix completion, universal rigidity and the Strong Arnold Property
This paper addresses the following three topics: positive semidefinite (psd) matrix completions, universal rigidity of frameworks, and the Strong Arnold Property (SAP). We show some strong connections among these topics, using semidefinite programming as unifying theme. Our main contribution is a sufficient condition for constructing partial psd matrices which admit a unique completion to a full psd matrix. Such partial matrices are an essential tool in the study of the Gram dimension \gd(G) of a graph G, a recently studied graph parameter related to the low psd matrix completion problem. Additionally, we derive an elementary proof of Connelly's sufficient condition for universal rigidity of tensegrity frameworks and we investigate the links between these two sufficient conditions. We also give a geometric characterization of psd matrices satisfying the Strong Arnold Property in terms of nondegeneracy of an associated semidefinite program, which we use to establish some links between the Gram dimension \gd(\cdot) and the Colin de Verdi\`ere type graph parameter ν=(⋅)
Positive semidefinite matrix completion, universal rigidity and the Strong Arnold Property
This paper addresses the following three topics: positive semidefinite (psd) matrix completions, universal rigidity of frameworks, and the Strong Arnold Property (SAP). We show some strong connections among these topics, using semidefinite programming as unifying theme. Our main contribution is a sufficient condition for constructing partial psd matrices which admit a unique completion to a full psd matrix. Such partial matrices are an essential tool in the study of the Gram dimension \gd(G) of a graph G, a recently studied graph parameter related to the low psd matrix completion problem. Additionally, we derive an elementary proof of Connelly's sufficient condition for universal rigidity of tensegrity frameworks and we investigate the links between these two sufficient conditions. We also give a geometric characterization of psd matrices satisfying the Strong Arnold Property in terms of nondegeneracy of an associated semidefinite program, which we use to establish some links between the Gram dimension \gd(\cdot) and the Colin de Verdi\`ere type graph parameter ν=(⋅)