Imai Laboratory Introduction
Studies on Discrete Mathematics
Matroids and Oriented Matroids
Matroids
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
Experimental
Source
Matroid
Database
dual
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
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)
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]
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.