Routine svd from Matlab 7.6.0.324 (R2008a) used to calculate the singular values.

Matrix properties (click for a legend) | |

number of rows | 601 |

number of columns | 676 |

structural full rank? | yes |

structural rank | 601 |

numerical rank | 540 |

dimension of the numerical null space | 136 |

numerical rank / min(size(A)) | 0.8985 |

Euclidean norm of A | 3.9995 |

calculated singular value # 540 | 0.40392 |

numerical rank defined using a tolerance max(size(A))*eps(norm(A)) = | 3.002e-013 |

calculated singular value # 541 | 2.6163e-015 |

gap in the singular values at the numerical rank: singular value # 540 / singular value # 541 | 1.5439e+014 |

calculated condition number | 1.0034e+016 |

condest | -2 |

nonzeros | 2,027 |

# of blocks from dmperm | 182 |

# strongly connected comp. | 1 |

entries not in dmperm blocks | 361 |

explicit zero entries | 0 |

nonzero pattern symmetry | 0% |

numeric value symmetry | 0% |

type | binary |

structure | rectangular |

Cholesky candidate? | no |

positive definite? | no |

author | S. Margulies |

editor | J.-G. Dumas |

date | 2008 |

kind | combinatorial problem |

2D/3D problem? | no |

SJid | 614 |

UFid | 2,164 |

Notes:

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/kneser_6_2_1.sms

Ordering statistics: | AMD |
METIS |
DMPERM+ |

nnz(V) for QR, upper bound nnz(L) for LU | 15,956 | 16,285 | 9,052 |

nnz(R) for QR, upper bound nnz(U) for LU | 11,007 | 11,493 | 6,080 |

*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.