Polytopes of Minimum Positive Semidefinite Rank
نویسندگان
چکیده
The positive semidefinite (psd) rank of a polytope is the smallest k for which the cone of k × k real symmetric psd matrices admits an affine slice that projects onto the polytope. In this paper we show that the psd rank of a polytope is at least the dimension of the polytope plus one, and we characterize those polytopes whose psd rank equals this lower bound.
منابع مشابه
Orthogonal Representations, Projective Rank, and Fractional Minimum Positive Semidefinite Rank: Connections and New Directions
Fractional minimum positive semidefinite rank is defined from r-fold faithful orthogonal representations and it is shown that the projective rank of any graph equals the fractional minimum positive semidefinite rank of its complement. An r-fold version of the traditional definition of minimum positive semidefinite rank of a graph using Hermitian matrices that fit the graph is also presented. Th...
متن کاملZero forcing parameters and minimum rank problems
Abstract. The zero forcing number Z(G), which is the minimum number of vertices in a zero forcing set of a 1 graph G, is used to study the maximum nullity/minimum rank of the family of symmetric matrices described by 2 G. It is shown that for a connected graph of order at least two, no vertex is in every zero forcing set. The positive 3 semidefinite zero forcing number Z+(G) is introduced, and ...
متن کاملComputing Positive Semidefinite Minimum Rank for Small Graphs
Abstract. The positive semidefinite minimum rank of a simple graph G is defined to be the smallest possible rank over all 1 positive semidefinite real symmetric matrices whose ijth entry (for i 6= j) is nonzero whenever {i, j} is an edge in G and is zero 2 otherwise. The computation of this parameter directly is difficult. However, there are a number of known bounding parameters 3 and technique...
متن کاملSupport-based lower bounds for the positive semidefinite rank of a nonnegative matrix
The positive semidefinite rank of a nonnegative (m×n)-matrix S is the minimum number q such that there exist positive semidefinite (q × q)-matrices A1, . . . , Am, B1, . . . , Bn such that S(k, l) = trA∗kBl. The most important lower bound technique on nonnegative rank only uses the zero/nonzero pattern of the matrix. We characterize the power of lower bounds on positive semidefinite rank based ...
متن کاملExploring the bounds on the positive semidefinite rank
The nonnegative and positive semidefinite (PSD-) ranks are closely connected to the nonnegative and positive semidefinite extension complexities of a polytope, which are the minimal dimensions of linear and SDP programs which represent this polytope. Though some exponential lower bounds on the nonnegative [FMP12] and PSD[LRS15] ranks has recently been proved for the slack matrices of some parti...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Discrete & Computational Geometry
دوره 50 شماره
صفحات -
تاریخ انتشار 2013