Matrix: JGD_Margulies/flower_5_4

Description: Combinatorial optimization as polynomial eqns, Susan Margulies, UC Davis

JGD_Margulies/flower_5_4 graph
(bipartite graph drawing)

JGD_Margulies/flower_5_4 dmperm of JGD_Margulies/flower_5_4

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  • download as a MATLAB mat-file, file size: 94 KB. Use UFget(2158) or UFget('JGD_Margulies/flower_5_4') in MATLAB.
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  • download in Rutherford/Boeing format, file size: 118 KB.

    Matrix properties
    number of rows5,226
    number of columns14,721
    structural full rank?yes
    structural rank5,226
    # of blocks from dmperm12
    # strongly connected comp.1
    explicit zero entries0
    nonzero pattern symmetry 0%
    numeric value symmetry 0%
    Cholesky candidate?no
    positive definite?no

    authorS. Margulies
    editorJ.-G. Dumas
    kindcombinatorial problem
    2D/3D problem?no


    Combinatorial optimization as polynomial eqns, Susan Margulies, UC Davis
    From Jean-Guillaume Dumas' Sparse Integer Matrix Collection,                      
    Expressing Combinatorial Optimization Problems by Systems of Polynomial 
    Equations and the Nullstellensatz                                       
    Authors: J.A. De Loera, J. Lee, Susan Margulies, S. Onn                 
    (Submitted on 5 Jun 2007)                                               
    Abstract: Systems of polynomial equations over the complex or real      
    numbers can be used to model combinatorial problems. In this way, a     
    combinatorial problem is feasible (e.g. a graph is 3-colorable,         
    hamiltonian, etc.) if and only if a related system of polynomial        
    equations has a solution. In the first part of this paper, we construct 
    new polynomial encodings for the problems of finding in a graph its     
    longest cycle, the largest planar subgraph, the edge-chromatic number,  
    or the largest k-colorable subgraph.  For an infeasible polynomial      
    system, the (complex) Hilbert Nullstellensatz gives a certificate that  
    the associated combinatorial problem is infeasible. Thus, unless P =    
    NP, there must exist an infinite sequence of infeasible instances of    
    each hard combinatorial problem for which the minimum degree of a       
    Hilbert Nullstellensatz certificate of the associated polynomial system 
    grows.  We show that the minimum-degree of a Nullstellensatz            
    certificate for the non-existence of a stable set of size greater than  
    the stability number of the graph is the stability number of the graph. 
    Moreover, such a certificate contains at least one term per stable set  
    of G. In contrast, for non-3- colorability, we found only graphs with   
    Nullstellensatz certificates of degree four.                            
    Filename in JGD collection: Margulies/flower_5_4.sms                    

    Ordering statistics:result
    nnz(V) for QR, upper bound nnz(L) for LU, with COLAMD26,107,872
    nnz(R) for QR, upper bound nnz(U) for LU, with COLAMD4,406,385

    SVD-based statistics:
    null space dimension0
    full numerical rank?yes

    singular values (MAT file):click here
    SVD method used:s = svd (full (A)) ;

    JGD_Margulies/flower_5_4 svd

    For a description of the statistics displayed above, click here.

    Maintained by Tim Davis, last updated 12-Mar-2014.
    Matrix pictures by cspy, a MATLAB function in the CSparse package.
    Matrix graphs by Yifan Hu, AT&T Labs Visualization Group.