Bounds for minimum semidefinite rank from superpositions and cutsets

Unitary matrix digraphs and minimum semidefinite rank

Matrices attaining the minimum semidefinite rank of a chordal graph

On minimum rank and zero forcing sets of a graph

Minimum rank and maximum eigenvalue multiplicity of symmetric tree sign patterns

We prove that the ordered subgraph number of a connected graph that has no duplicate vertices is at most three if and only if the complement does not contain a cycle on four vertices. The duality between zero forcing and ordered subgraphs then provides a complementary characterization for positive semidefinite zero forcing. We also provide some necessary conditions for when the minimum semidefinite rank can be computed using tree size.

We construct a bipartite generalization of Alon and Szegedy's nearly orthogonal vectors, thereby obtaining strong bounds for several extremal problems involving the Lovász theta function, vector chromatic number, minimum semidefinite rank, nonnegative rank, and extension complexity of polytopes. In particular, we answer a question from our previous work together with Letzter and Sudakov, while also addressing a question of Hrubeš and of Kwan, Sauermann, and Zhao. Along the way, we derive a couple of general lower bounds for the vector chromatic number which may be of independent interest.

Decompositions of minimum rank matrices

Minimum rank problems

Minimum rank of outerplanar graphs

Matrix completion problems studies the partial matrices, that is, a rectangular matrix some of whose entries are specified, and the remainder entries are free variables of some indicated set. By a completion of a partial matrix we consider a specification of the free variables obtaining a conventional matrix. The basic type of these problems try to obtain conditions for the existence of a completion for a given partial matrix in a class of interest.In this work, we study the minimal rank completion problem when the partial matrix P has the specified entries equal to zero, and the remaining entries are positive real numbers. By a graph theoretic approach we introduce some approximations to the question. Furthermore, we obtain completions for some classes of positive pattern matrices with minimal rank.

The minimum rank of a graph G is the minimum of the ranks of all symmetric adjacency matrices of G. We present a new combinatorial bound for the minimum rank of an arbitrary graph G based on enumerating certain subsets of vertices of G satisfying matroid theoretic properties. We also present some computational and theoretical challenges associated with computing the minimum rank. This includes a conjecture that this bound on the minimum rank actually holds with equality for all graphs. History: This “Challenge” paper was invited by the Editor in Chief and based on the topics raised by the author at his plenary address at the 2022 INFORMS Computing Society Conference in Tampa, Florida. Funding: This work was supported by the National Science Foundation [Grant DMS-1720225]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2022.1219 .

On the difference between the maximum multiplicity and path cover number for tree-like graphs

On the graph complement conjecture for minimum semidefinite rank

This paper is based on a recent work by Kojima which extended sums of squares relaxations of polynomial optimization problems to polynomial semidefinite programs. Let $$\varepsilon$$ and $$\varepsilon_{+}$$ be a finite dimensional real vector space and a symmetric cone embedded in $$\varepsilon$$; examples of $$\varepsilon$$ and $$\varepsilon_{+}$$ include a pair of the N-dimensional Euclidean space and its nonnegative orthant, a pair of the N-dimensional Euclidean space and N-dimensional second-order cones, and a pair of the space of m × m real symmetric (or complex Hermitian) matrices and the cone of their positive semidefinite matrices. Sums of squares relaxations are further extended to a polynomial optimization problem over $$\varepsilon_{+}$$, i.e., a minimization of a real valued polynomial a(x) in the n-dimensional real variable vector x over a compact feasible region $$\{ {\bf x} : b({\bf x}) \in \varepsilon_{+}\}$$, where b(x) denotes an $$\varepsilon$$- valued polynomial in x. It is shown under a certain moderate assumption on the $$\varepsilon$$-valued polynomial b(x) that optimal values of a sequence of sums of squares relaxations of the problem, which are converted into a sequence of semidefinite programs when they are numerically solved, converge to the optimal value of the problem.

From Boolean to sign pattern matrices