Imai Laboratory Introduction Studies on Discrete Mathematics Matroids

Imai Laboratory Introduction Studies on Discrete Mathematics Matroids and Oriented Matroids (OM) Realizable OM Orientable Matroids The realizability problem of oriented matroids Orientability of Matroids Is there the coresponding oriented matroids? Conditions of orientable matroids: Mnëv’s universality theorem 3 -term Grassman-Plücker(GP) relations Is there the coresponding vector configurations? Existential theory of the reals Does this system has a solution over the real numbers? equivalent logical formula SDP relaxation Give realizations by polynomial optimization Solve as SAT problem joint work with D. Bremner Result: determined orientability of rank 3 matroids on up to 12 elements rank 4 matroids on up to 9 elements Experimental Source Matroid Database Enumeration of Matroids We want larger database Construct lager database! First Database by Blackburn, Crapo and Higgs (1973) Effecient algorithm by the reverse search accomplishment: Encoding of matroids Exhaustive enumeration of rank 4 matroids on 10 elements Based on enconding of OMs (4, 886, 380, 924 matroids) dual Non-realizability certificates by polynomial optimization Result: 8 elements and rank 4 realizable Solvability sequence Not-isolated elements Still unknown 171344 5321 Apply POP & generalized mutation +3432 graphs Experimental Source non- realizable BFP 3968 non-Euclidean 3462 non-HK* 1382 440 Non-realizability certificates by SDP Oriented Matroid Database Work by L. Finschi and K. Fukuda by Finschi and Fukuda Reference: [1] References Reference: [2, 3] [1] Yoshitake Matsumoto, Sonoko Moriyama, Hiroshi Imai: Enumeration of Matroids by Reverse Search and Its Applications, Kyoto. CGGT 2007, 2007/6/11 -15. [2] Hiroyuki Miyata, Sonoko Moriyama, Hiroshi Imai: Determining the non-realizability of oriented matroids by semidefinite programming, Kyoto. CGGT 2007, 2007/6/11 -15. [3] Hiroki Nakayama, Sonoko Moriyama and Komei Fukuda: Realizations of non-uniform oriented matroids using generalized mutation graphs, in Proc. of the 5 th Hungarian-Japanese Symposium on Discrete Mathematics and Its Applications, pp. 242 -251, 2007.