A new graph parameter related to bounded rank positive semidefinite matrix completions

@article{Laurent2014ANG,
  title={A new graph parameter related to bounded rank positive semidefinite matrix completions},
  author={Monique Laurent and Antonios Varvitsiotis},
  journal={Mathematical Programming},
  year={2014},
  volume={145},
  pages={291-325}
}
The Gram dimension $$\mathrm{gd}(G)$$ of a graph $$G$$ is the smallest integer $$k\ge 1$$ such that any partial real symmetric matrix, whose entries are specified on the diagonal and at the off-diagonal positions corresponding to edges of $$G$$, can be completed to a positive semidefinite matrix of rank at most $$k$$ (assuming a positive semidefinite completion exists). For any fixed $$k$$ the class of graphs satisfying $$\mathrm{gd}(G) \le k$$ is minor closed, hence it can be characterized by… 
On bounded rank positive semidefinite matrix completions of extreme partial correlation matrices
TLDR
An upper bound foregd(G) is shown in terms of a new tree-width-like parameter $\sla(G), defined as the smallest $r$ for which $G$ is a minor of the strong product of a tree and $K_r$ is shown.
Universal Completability, Least Eigenvalue Frameworks, and Vector Colorings
TLDR
This work identifies a sufficient condition for showing that a graph is UVC based on the universal completability of an associated framework and proves that Kneser and q-Kneser graphs are UVC.
Unavoidable Minors for Graphs with Large $$\ell _p$$ ℓ p -Dimension
TLDR
This paper characterize the minor-closed graph classes C with bounded ℓ p -dimension, and gives a simple proof that C has boundedℓ 2 -dimension if and only if $$\mathscr {C}$$C has bounded treewidth.
The Signed Positive Semidefinite Matrix Completion Problem for Odd-$K_4$ Minor Free Signed Graphs
We give a signed generalization of Laurent's theorem that characterizes feasible positive semidefinite matrix completion problems in terms of metric polytopes. Based on this result, we give a
Do Sums of Squares Dream of Free Resolutions?
TLDR
The results allow us to generalize the work of Blekherman-Smith-Velasco on equality of nonnegative polynomials and sums of squares from irreducible varieties to reduced schemes and to classify all spectrahedral cones with only rank one extreme rays.
The Gram Dimension of a Graph
TLDR
It is shown that a graph has Gram dimension at most 4 if and only if it does not have K5 and K2, 2,2 as minors, which implies the characterization of 3-realizable graphs of Belk and Connelly.
Sparse semidefinite programs with guaranteed near-linear time complexity via dualized clique tree conversion
TLDR
By dualizing the clique tree conversion, the coupling due to the overlap constraints is guaranteed to be sparse over dense blocks, with a block sparsity pattern that coincides with the adjacency matrix of a tree.
Combinatorial Conditions for the Unique Completability of Low-Rank Matrices
TLDR
This work considers the problems of completing a low-rank positive semidefinite square matrix or aLow-rank rectangular matrix from a given subset of their entries, and studies the local and global uniqueness of such completions by analyzing the structure of the graphs determined by the positions of the known entries.
Exact semidefinite formulations for a class of (random and non-random) nonconvex quadratic programs
TLDR
This work studies a class of quadratically constrained quadratic programs (QCQPs), called diagonal QCQPs, and provides a sufficient condition on the problem data guaranteeing that the basic Shor semidefinite relaxation is exact, the first result establishing the exactness of the semideFinite relaxation for random general QCZPs.
...
...

References

SHOWING 1-10 OF 50 REFERENCES
Polynomial Instances of the Positive Semidefinite and Euclidean Distance Matrix Completion Problems
  • M. Laurent
  • Mathematics
    SIAM J. Matrix Anal. Appl.
  • 2001
TLDR
The matrix completion problem can be solved in polynomial time in the real number model for the class of graphs containing no homeomorph K4 and the completion problem is polynomially solvable for a class of graph including wheels of fixed length.
Variants on the minimum rank problem: A survey II
TLDR
This paper surveys the many developments on the (standard) minimum rank problem and its variants since the survey paper, and positive semidefinite minimum rank, zero forcing parameters, and minimum rank problems for patterns are discussed.
The Gram Dimension of a Graph
TLDR
It is shown that a graph has Gram dimension at most 4 if and only if it does not have K5 and K2, 2,2 as minors, which implies the characterization of 3-realizable graphs of Belk and Connelly.
Complexity of the positive semidefinite matrix completion problem with a rank constraint
We consider the decision problem asking whether a partial rational symmetric matrix with an all-ones diagonal can be completed to a full positive semidefinite matrix of rank at most k. We show that
Embedded in the Shadow of the Separator
TLDR
This work proves that, for any separator in the graph, at least one of the two separated node sets is embedded in the shadow (with the origin being the light source) of the convex hull of the separator.
The rotational dimension of a graph
TLDR
The rotational dimension of G is defined to be the minimal dimension k so that for all choices of lengths and weights an optimal solution can be found in ℝk and it is shown that this is a minor monotone graph parameter.
The real positive semidefinite completion problem for series-parallel graphs
Rounding Two and Three Dimensional Solutions of the SDP Relaxation of MAX CUT
TLDR
An improved rounding procedure for SDP solutions that lie in ℝ3 with a performance ratio of about 0.8818 is presented, which resolves an open problem posed by Feige and Schechtman [STOC'01].
A Remark on the Rank of Positive Semidefinite Matrices Subject to Affine Constraints
TLDR
A short geometric proof of this result is given, which is used to improve a bound on realizability of weighted graphs as graphs of distances between points in Euclidean space, and its relation to theorems of Bohnenblust, Friedland and Loewy, and Au-Yeung and Poon.
...
...