1 / 1

Imai Laboratory Introduction Studies on Discrete Mathematics

Imai Laboratory Introduction Studies on Discrete Mathematics. Matroids and Oriented Matroids. Matroids. Oriented Matroids (OM). Orientable Matroids. Realizable OM. The realizability problem of oriented matroids. Orientability of Matroids. Is there the coresponding oriented matroids?.

madge
Download Presentation

Imai Laboratory Introduction Studies on Discrete Mathematics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Imai Laboratory Introduction Studies on Discrete Mathematics Matroids and Oriented Matroids Matroids Oriented Matroids (OM) Orientable Matroids Realizable OM The realizability problem of oriented matroids Orientability of Matroids Is there the corespondingoriented matroids? Is there the coresponding vector configurations? Conditions of orientable matroids:3-term Grassman-Plücker(GP) relations Mnëv’s universality theorem 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 Non-realizability certificates by polynomial optimization dual 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 Result: 8 elements and rank 4 realizable non- realizable Still unknown BFP 3968 Solvability sequence Experimental Source Enumeration of Matroids Not-isolated elements non-Euclidean 3462 171344 5321 Matroid Database non-HK* We want larger database 1382 Apply POP & generalized mutation graphs 440 Construct lager database! +3432 First Database byBlackburn, Crapo and Higgs (1973) Effecient algorithm by the reverse search Experimental Source Non-realizability certificates by SDP accomplishment: Exhaustive enumeration of rank 4 matroids on 10 elements (4,886,380,924 matroids) Oriented Matroid Database Encoding of matroids Based on enconding of OMs by Finschi and Fukuda Work by L. Finschi and K. Fukuda Reference: [1] Reference: [2,3] References [1] Yoshitake Matsumoto, Sonoko Moriyama, Hiroshi Imai: Enumeration of Matroids by Reverse Search and Its Applications, KyotoCGGT2007, 2007/6/11-15. [2] Hiroyuki Miyata, Sonoko Moriyama, Hiroshi Imai: Determining the non-realizability of oriented matroids by semidefinite programming, KyotoCGGT2007, 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 5th Hungarian-Japanese Symposium on Discrete Mathematics and Its Applications, pp.242-251, 2007.

More Related