**Matrix: JGD_Margulies/cat_ears_2_1**

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

(bipartite graph drawing) | (graph drawing of A+A') |

Matrix properties | |

number of rows | 85 |

number of columns | 85 |

nonzeros | 254 |

structural full rank? | no |

structural rank | 82 |

# of blocks from dmperm | 15 |

# strongly connected comp. | 1 |

explicit zero entries | 0 |

nonzero pattern symmetry | 3% |

numeric value symmetry | 3% |

type | binary |

structure | unsymmetric |

Cholesky candidate? | no |

positive definite? | no |

author | S. Margulies |

editor | J.-G. Dumas |

date | 2008 |

kind | combinatorial problem |

2D/3D problem? | no |

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/cat_ears_2_1.sms

Ordering statistics: | result |

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 |

SVD-based statistics: | |

norm(A) | 3.42573 |

min(svd(A)) | 7.71606e-18 |

cond(A) | 4.43974e+17 |

rank(A) | 74 |

sprank(A)-rank(A) | 8 |

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)) ; |

status: | ok |

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.
*