• SJSU Singular Matrix Database
  • Matrix group: JGD_Margulies
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  • Matrix: JGD_Margulies/wheel_4_1
  • Description: Combinatorial optimization as polynomial eqns, Susan Margulies, UC Davis
  • download as a MATLAB mat-file, file size: 3 KB. Use SJget(555) or SJget('JGD_Margulies/wheel_4_1') in MATLAB.
  • download in Matrix Market format, file size: 2 KB.
  • download in Rutherford/Boeing format, file size: 2 KB.


    Routine svd from Matlab (R2008a) used to calculate the singular values.


    dmperm of JGD_Margulies/wheel_4_1

    Matrix properties (click for a legend)  
    number of rows36
    number of columns41
    structural full rank?yes
    structural rank36
    numerical rank 32
    dimension of the numerical null space9
    numerical rank / min(size(A))0.88889
    Euclidean norm of A 3.3697
    calculated singular value # 320.67271
    numerical rank defined using a tolerance
    max(size(A))*eps(norm(A)) =
    calculated singular value # 336.3471e-016
    gap in the singular values at the numerical rank:
    singular value # 32 / singular value # 33
    calculated condition number1.5646e+016
    # of blocks from dmperm2
    # strongly connected comp.1
    entries not in dmperm blocks1
    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/wheel_4_1.sms                     

    Ordering statistics:AMD METIS DMPERM+
    nnz(V) for QR, upper bound nnz(L) for LU204 265 208
    nnz(R) for QR, upper bound nnz(U) for LU257 295 257

    Maintained by Leslie Foster, last updated 24-Apr-2009.

    Entries 5 through 14 in the table of matrix properties and the singular
    value plot were created using SJsingular code. The other plots
    and statistics are produced using utilities from the SuiteSparse package.
    Matrix color plot pictures by cspy, a MATLAB function in the CSparse package.