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

Monique Laurent, Antonios Varvitsiotis
2012 arXiv   pre-print
to edges of G, can be completed to a positive semidefinite matrix of rank at most k (assuming a positive semidefinite completion exists).  ...  We also show some close connections to Euclidean realizations of graphs and to the graph parameter ν^=(G) of H03.  ...  -Nagy for useful discussions and A. Schrijver for his suggestions for the proof of Theorem 5.  ... 
arXiv:1204.0734v1 fatcat:y7eue5ijkrejzg2j7dz5uy7aue

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

Monique Laurent, Antonios Varvitsiotis
2013 Mathematical programming  
to edges of G, can be completed to a positive semidefinite matrix of rank at most k (assuming a positive semidefinite completion exists).  ...  We also show some close connections to Euclidean realizations of graphs and to the graph parameter ν = (G) of [21] .  ...  -Nagy for useful discussions and A. Schrijver for his suggestions for the proof of Theorem 5.  ... 
doi:10.1007/s10107-013-0648-x fatcat:fcsomo7egnh6dpqmm76x2slqmi

Low-rank solutions of matrix inequalities with applications to polynomial optimization and matrix completion problems

Ramtin Madani, Ghazal Fazelnia, Somayeh Sojoudi, Javad Lavaei
2014 53rd IEEE Conference on Decision and Control  
We develop a theory relating the rank of the minimum-rank solution of the LMI problem to the sparsity of its underlying graph.  ...  This paper is concerned with the problem of finding a low-rank solution of an arbitrary sparse linear matrix inequality (LMI). To this end, we map the sparsity of the LMI problem into a graph.  ...  Low-rank Positive Semidefinite Matrix Completion The LMI problem (1) encapsulates the low-rank positive semidefinite matrix completion problem, which is described below.  ... 
doi:10.1109/cdc.2014.7040064 dblp:conf/cdc/MadaniFSL14 fatcat:muahfrpfijcb5aasa7rzghcn2a

On tropical and nonnegative factorization ranks of band matrices [article]

Yaroslav Shitov
2017 arXiv   pre-print
We give a survey of what is known on the algorithmic complexity of Boolean, fuzzy, tropical, nonnegative, and positive semidefinite factorizations, and we examine the behavior of the corresponding rank  ...  We show that the Boolean, fuzzy, and tropical versions of matrix factorization become polynomial time solvable when restricted to this class of matrices, and we also show that the nonnegative rank of a  ...  Problem 14 can be related to the question asked in [11] . Namely, is it NP-hard to decide if the positive semidefinite rank of a n × n matrix equals n?  ... 
arXiv:1710.02072v1 fatcat:j44tc2ycvbazjmbci26lq4phui

The Gram dimension of a graph [article]

Monique Laurent, Antonios Varvitsiotis
2012 arXiv   pre-print
The Gram dimension (G) of a graph is the smallest integer k > 1 such that, for every assignment of unit vectors to the nodes of the graph, there exists another assignment of unit vectors lying in ^k, having  ...  We show that a graph has Gram dimension at most 4 if and only if it does not have K_5 and K_2,2,2 as minors. We also show some close connections to the notion of d-realizability of graphs.  ...  -Nagy for useful discussions and A. Schrijver for his suggestions for the proof of Theorem 5.  ... 
arXiv:1112.5960v2 fatcat:i72t263klbeg5jrlqdcqahiwru

Maximum Likelihood Threshold and Generic Completion Rank of Graphs [article]

Grigoriy Blekherman, Rainer Sinn
2017 arXiv   pre-print
On the other hand we determine both invariants for complete bipartite graphs K_m,n and show that for some choices of m and n the two parameters may be quite far apart.  ...  In particular, this gives the first examples of graphs on which the maximum likelihood threshold and the generic completion rank do not agree.  ...  In the end of this paper, we relate the generic completion rank of bipartite graphs to the generic completion rank for the general matrix completion problem: Given a partially specified m × n matrix, we  ... 
arXiv:1703.07849v2 fatcat:36ym77ebgjgadibh3y5gv5jude

Using Algebraic Properties of Minimal Idempotents for Exhaustive Computer Generation of Association Schemes

K. Coolsaet, J. Degraer
2008 Electronic Journal of Combinatorics  
Central to our success is the use of two algebraic constraints based on properties of the minimal idempotents $E_{i}$ of these association schemes : the fact that they are positive semidefinite and that  ...  $(126,50,13,24)$ graph and some new examples of antipodal distance regular graphs.  ...  Indeed, any principal submatrix of a positive semidefinite matrix must again be positive semidefinite, and any principal submatrix of a matrix must have a rank which is at most the rank of the original  ... 
doi:10.37236/754 fatcat:ddilpk2ej5dybel2q7z4gcnbzm

On the graph complement conjecture for minimum rank

Francesco Barioli, Wayne Barrett, Shaun M. Fallat, H. Tracy Hall, Leslie Hogben, Hein van der Holst
2012 Linear Algebra and its Applications  
Related results for the graph complement conjecture (and the positive semidefinite version) for joins of graphs are also established.  ...  This conjecture asks for an upper bound on the sum of the minimum rank of a graph and the minimum rank of its complement, and may be classified as a Nordhaus-Gaddum type problem involving the graph parameter  ...  Acknowledgments The authors thank Sebastian Cioaba for suggesting we examine connections between the Graph Complement Conjecture and other types of Nordhaus-Gaddum graph parameter problems.  ... 
doi:10.1016/j.laa.2010.12.024 fatcat:42izy2n34jga3jded7t7dg7pva

Positive semidefinite maximum nullity and zero forcing number

Travis Peters
2012 The Electronic Journal of Linear Algebra  
Since any positive semidefinite matrix A can be written as X T X for some X ∈ M n (R) with rank A = rank X, every positive semidefinite matrix is a Gram Matrix.  ...  We use this result to determine the vertex connectivity of the Cartesian product of a complete graph and a path, thereby providing a lower bound for the maximum positive semidefinite nullity.  ... 
doi:10.13001/1081-3810.1559 fatcat:yxecfsv6ube2lmnhzddsadt4lq

Matrices with high completely positive semidefinite rank

Sander Gribling, David de Laat, Monique Laurent
2017 Linear Algebra and its Applications  
We also exhibit a class of completely positive matrices with quadratic (in terms of the matrix size) completely positive rank, but with linear completely positive semidefinite rank, and we make a connection  ...  The smallest such d is called the (complex) completely positive semidefinite rank of M, and it is an open question whether there exists an upper bound on this number as a function of the matrix size.  ...  We are grateful to an anonymous referee for his/her careful reading and helpful comments, and for bringing the works [15, 27] to our attention.  ... 
doi:10.1016/j.laa.2016.10.015 fatcat:h6mf372z6needfeaznss2ffcqm

Finding Low-rank Solutions of Sparse Linear Matrix Inequalities using Convex Optimization

Ramtin Madani, Somayeh Sojoudi, Ghazal Fazelnia, Javad Lavaei
2017 SIAM Journal on Optimization  
We develop a theory relating the rank of the minimum-rank solution of the LMI problem to the sparsity of its underlying graph.  ...  This paper is concerned with the problem of finding a low-rank solution of an arbitrary sparse linear matrix inequality (LMI). To this end, we map the sparsity of the LMI problem into a graph.  ...  The authors would like to thank Professors Daniel Bienstock and Vineet Goyal of Columbia University for several fruitful discussions.  ... 
doi:10.1137/14099379x fatcat:u2b7c2chvvfk7kuemvdy5fnpva

Conic approach to quantum graph parameters using linear optimization over the completely positive semidefinite cone [article]

Monique Laurent, Teresa Piovesan
2015 arXiv   pre-print
We investigate the completely positive semidefinite cone CS_+^n, a new matrix cone consisting of all n× n matrices that admit a Gram representation by positive semidefinite matrices (of any size).  ...  We use this new cone to model quantum analogues of the classical independence and chromatic graph parameters α(G) and χ(G), which are roughly obtained by allowing variables to be positive semidefinite  ...  In particular, we are grateful to one referee for suggesting the result of Lemma 4.4 which permitted to correct an error in the proof of Proposition 4.5.  ... 
arXiv:1312.6643v6 fatcat:vcpn7hmhqrcgpae3zz3ox2ytaa

Conic Approach to Quantum Graph Parameters Using Linear Optimization Over the Completely Positive Semidefinite Cone

Monique Laurent, Teresa Piovesan
2015 SIAM Journal on Optimization  
We investigate the completely positive semidefinite cone CS n + , a new matrix cone consisting of all n × n matrices that admit a Gram representation by positive semidefinite matrices (of any size).  ...  We use this new cone to model quantum analogues of the classical independence and chromatic graph parameters α(G) and χ(G), which are roughly obtained by allowing variables to be positive semidefinite  ...  In particular, we are grateful to one referee for suggesting the result of Lemma 4.4 which permitted to correct an error in the proof of Proposition 4.5.  ... 
doi:10.1137/14097865x fatcat:c4j45vljn5eg5btyxqxtgj33me

CONNECTION MATRICES [chapter]

László Lovász
2007 Combinatorics, Complexity, and Chance  
Acknowledgement I am grateful to the anonymous referee for pointing out the construction in Example 2.11.  ...  The property that M (f, k) is positive semidefinite is related to the "reflection positivity" property in statistical physics, and we'll call a graph parameter reflection positive if M (f, k) is positive  ...  For the chromatic polynomial chr(G, q), it is easy to get an upper bound on the rank of the connection matrix M plan (chr(., q), k).  ... 
doi:10.1093/acprof:oso/9780198571278.003.0012 fatcat:h33lwmporjdwhkfkcdt4ztsq7e

Reflection positivity, rank connectivity, and homomorphism of graphs

Michael Freedman, László Lovász, Alexander Schrijver
2006 Journal of The American Mathematical Society  
Acknowledgement We are indebted to Christian Borgs, Jennifer Chayes, Monique Laurent, Miki Simonovits, Vera T.  ...  Sós, Balázs Szegedy, Gábor Tardos and Kati Vesztergombi for many valuable discussions and suggestions on the topic of graph homomorphisms.  ...  matrix M (f, k) is positive semidefinite and has rank 1.  ... 
doi:10.1090/s0894-0347-06-00529-7 fatcat:yalpe4lumrgxbcvn3wdhrc7lz4
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