Matrix: LPnetlib/lp_qap12

Description: Netlib LP problem qap12: minimize c'*x, where Ax=b, lo<=x<=hi

LPnetlib/lp_qap12 graph
(bipartite graph drawing)


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    Matrix properties
    number of rows3,192
    number of columns8,856
    structural full rank?yes
    structural rank3,192
    # of blocks from dmperm1
    # strongly connected comp.1
    explicit zero entries0
    nonzero pattern symmetry 0%
    numeric value symmetry 0%
    Cholesky candidate?no
    positive definite?no

    authorT. Johnson
    editorR. Bixby, M. Saltzman, T. Johnson
    kindlinear programming problem
    2D/3D problem?no

    Additional fieldssize and type
    bfull 3192-by-1
    cfull 8856-by-1
    lofull 8856-by-1
    hifull 8856-by-1
    z0full 1-by-1


    A Netlib LP problem, in lp/generators/qap.  For more information          
    send email to with the message:                           
    	 send index from lp                                                      
    	 send readme from lp/data                                                
    	 send readme from lp/generators/qap                                      
    This copy of QAP12 was created by the QAP generator program,              
    on an Sun UltraSparc, on May 15, 1997.                                    
    The following are relevant excerpts from lp/data/readme (by David M. Gay):
    The column and nonzero counts in the PROBLEM SUMMARY TABLE below exclude  
    slack and surplus columns and the right-hand side vector, but include     
    the cost row.  We have omitted other free rows and all but the first      
    right-hand side vector, as noted below.  The byte count is for the        
    MPS compressed file; it includes a newline character at the end of each   
    line.  These files start with a blank initial line intended to prevent    
    mail programs from discarding any of the data.  The BR column indicates   
    whether a problem has bounds or ranges:  B stands for "has bounds", R     
    for "has ranges".                                                         
    The optimal value is from MINOS version 5.3 (of Sept. 1988)               
    running on a VAX with default options.                                    
                           PROBLEM SUMMARY TABLE                              
    Name       Rows   Cols   Nonzeros    Bytes  BR      Optimal Value         
    QAP12      3193   8856    44244 (see NOTES)       5.2289435056E+02        
    Problems QAP8, QAP12, and QAP15 are from a generator by Terri             
    Johnson (communicated by a combination of Bob Bixby, Matt Saltzman, and   
    Terri Johnson).                                                           
    Source for Terri Johnson's generator and input data                       
    for producing MPS files for QAP8, QAP12, and QAP15 appear in directory    
    Added to Netlib on 12 April 1996.                                         
    The following are relevant excerpts from lp/generators/qap/readme         
    (by Terri Johnson):                                                       
            The Quadratic Assignment Problem (Problem QAP) is a specially-    
    structured zero-one quadratic programming problem.  While having          
    received considerable attention since its introduction into the           
    literature over 30 years ago, and while many applications exist in        
    various disciplines, this problem has resisted exact solution             
    procedures.  Only for smaller-size problems can optimal solutions be      
    obtained and verified.  The solution strategies for Problem QAP           
    developed by Johnson (Ph.D.  dissertation, Clemson University, 1992)      
    and Adams and Johnson (Improved Linear Programming-based Lower Bounds     
    for the Quadratic Assignment Problem, DIMACS:  Quadratic Assignment       
    and Related Problems, Vol. 16 (1994), 43-75) are based on a new,          
    equivalent, mixed- integer linear reformulation, Problem LP.              
            The traditional , nonlinear formulation of Problem QAP has a      
    quadratic objective function, 2m constraints and m^2 binary variables.    
    The linearized version of concern, Problem LP, on the other hand, has     
    2m^2(m-1) + m^2(m-1)^2/2 + 2m constraints, in addition to non-            
    negativity restrictions on all the variables, and m^2 binary variables    
    and m^2(m-1)^2 continuous variables.  The continuous relaxation of        
    Problem LP, obtained by omitting the x binary restrictions, possesses     
    a special block diagonal structure which readily lends itself to          
    decomposition techniques.  However, the inherent degeneracy makes this    
    a formidable program for problems as small in size as m=15 to 20.  A      
    smaller reformulation, which reduces the number of constraints and        
    variables each by m^2(m-1)^2/2, can be obtained via an appropriate        
    substitution of variables, but such a substitution forfeits the           
    problem structure.  It has been amply demonstrated that this              
    formulation serves as a unifying and dominating entity with respect to    
    the different linear reformulations of Problem QAP, as well as with       
    respect to a variety of bounding procedures.  Consequently, the           
    ability to quickly solve this linear formulation holds the promise of     
    being able to solve larger-sized QAP's.                                   
            Provided here is Fortran source, newlp.f, for a program that      
    generates MPS files for the linearized QAP with the substitution of       
    variables.  Under the assumption that the test problem is symmetric,      
    the generator reads the problem size, m, and an mxm matrix with the       
    original distances in the upper half of the matrix and the original       
    flows in the lower half of the matrix.  All diagonal entries are 0.       
    Using this input, the generator program computes the objective            
    function coefficients for the quadratic terms, and automatically          
    computes the constraints.  The objective function is assumed to           
    contain no linear terms since such values can be easily incorporated      
    into the quadratic terms.                                                 
            Input files qap8.dat, qap12.dat, and qap15.dat cause the          
    generator program to emit MPS files for well-known test problems of       
    Nugent, C.E., T.E. Vollmann, and J. Ruml, An Experimental Comparison      
    of Techniques for the Assignment of Facilities to Locations,              
    Operations Research, Vol. 16, No. 1 (1968), 150-173, of sizes m=8, 12,    
    and 15 for the linearization.                                             
    PROBLEM:  M = 12		No. of Variables	No. of Constraints                     
    	QAP				144			  24                                                        
    	LP (with substitution)	       8856			3192                                
    	Optimal value: 5.2289435056e+2                                           
    For more information, please contact Terri Johnson at:                                                                  

    Ordering statistics:result
    nnz(V) for QR, upper bound nnz(L) for LU, with COLAMD13,738,096
    nnz(R) for QR, upper bound nnz(U) for LU, with COLAMD2,407,592

    SVD-based statistics:
    null space dimension398
    full numerical rank?no
    singular value gap2.72779e+14

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

    LPnetlib/lp_qap12 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.