Forbidden minor characterizations for low-rank optimal solutions to semidefinite programs over the elliptope
Nagy,M. Laurent,M. Varvitsiotis,A.
Nagy,M.
Laurent,M.
Varvitsiotis,A.
Abstract
We study a new geometric graph parameter egd(G), defined as the smallest integer r⩾1 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 G, can be completed to a matrix in the convex hull of correlation matrices of rank at most r. 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)⩽r if and only if the rank-r Grothendieck constant of G is equal to 1. We show that the parameter egd(G) is minor monotone, we identify several classes of forbidden minors for egd(G)⩽r and we give the full characterization for the case r=2. We also show an upper bound for egd(G) in terms of a new tree-width-like parameter la⊠(G), defined as the smallest r for which G is a minor of the strong product of a tree and Kr. We show that, for any 2-connected graph G≠K3,3 on at least 6 nodes, egd(G)⩽2 if and only if la⊠(G)⩽2.
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2014-09
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Research Projects
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Nagy, M, Laurent, M & Varvitsiotis, A 2014, 'Forbidden minor characterizations for low-rank optimal solutions to semidefinite programs over the elliptope', Journal of Combinatorial Theory Series B, vol. 108, no. September 2014, pp. 40-80. https://doi.org/10.1016/j.jctb.2014.02.011
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info:eu-repo/semantics/restrictedAccess