Description: Combinatorial optimization as polynomial eqns, Susan Margulies, UC Davis
|(bipartite graph drawing)||(graph drawing of A+A')|
|number of rows||85|
|number of columns||85|
|structural full rank?||no|
|# of blocks from dmperm||15|
|# strongly connected comp.||1|
|explicit zero entries||0|
|nonzero pattern symmetry||3%|
|numeric value symmetry||3%|
Combinatorial optimization as polynomial eqns, Susan Margulies, UC Davis From Jean-Guillaume Dumas' Sparse Integer Matrix Collection, http://ljk.imag.fr/membres/Jean-Guillaume.Dumas/simc.html http://arxiv.org/abs/0706.0578 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/cat_ears_2_1.sms
|nnz(chol(P*(A+A'+s*I)*P')) with AMD||1,045|
|Cholesky flop count||1.8e+04|
|nnz(L+U), no partial pivoting, with AMD||2,005|
|nnz(V) for QR, upper bound nnz(L) for LU, with COLAMD||581|
|nnz(R) for QR, upper bound nnz(U) for LU, with COLAMD||962|
|null space dimension||11|
|full numerical rank?||no|
|singular value gap||3.70665e+14|
|singular values (MAT file):||click here|
|SVD method used:||s = svd (full (A)) ;|
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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.