The Flow Lattice of Oriented Matroids Winfried Hochstättler, Robert Nickel Mathematical Foundations of Computer Science Department of Mathematics Brandenburg Technical University Cottbus
Outline Circuits and flows Reorientation and geometry An approach to define a flow number of an oriented matroid The flow lattice of regular oriented matroids of rank 3 oriented matroids of uniform oriented matroids Outlook
Circuits and Flows in Digraphs Directed circuit : Circular flow Flow number: 1 -1 2 -2 1 -1 1 4 3 5 7 2 8 6 9
Points in the Space Radon’s Theorem (1952): Let pairwise different. For all with exists a partition (Radon partition), so that Such a partition implies a signing of the elements 1 2 3 S T 5 4 6 7 Generalization of circuits in loop-less digraphs
Reorientation and Geometry digraphs are reoriented by flipping edges for point configurations we need some projective geometry: put the points on the projective sphere reorientation of an element is done by replacing the point by its double on the opposite half sphere a projective transformation then defines a new equator so that all points are on one half sphere (see Grünbaum – Convex Polytopes) 1 2 3 4 5 Circuits: 1234:+--+0 1235:+--0+ 1245:+-0-+ 1345:+0+-+ 2345:0++-+ Circuits reor.: 1234:+-++0 1235:+-+0+ 1345:+0--+ 2345:0+--+
Circuits and Flows of an Oriented Matroid Let be a family of signed subsets of a finite set that satisfies the following conditions: Then is the set of signed circuits of an oriented matroid A flow in is an integer combination of signed characteristic vectors of circuits:
A Flow Number for Oriented Matroids Goddyn, Tarsi, Zhang 1998: Let be the set of co-circuits of and the set of all reorientations. Then the oriented flow number is defined as For graphic matroids equal to the circular flow number of the graph (involves Hoffman’s Circulation Lemma 1960: for each bond in the digraph) Rank 3: (M. Edmonds, McNulty 2004) General case (co-connected): (Goddyn, Hliněný, Hochstättler)
Not a Matroid Invariant Different orientations of the same underlying matroid (e. g. ) can lead to a different oriented flow number - + - +
The Flow Lattice The flow lattice of an oriented matroid is defined as We define a flow number of analog to the flow number of a digraph What is the dimension of ? Does have a short characterization? Does contain a basis of ? Determine the flow number!
Regular Oriented Matroids (Digraphs and more) Concerning the dimension of we have is regular The elementary circuits to a basis of form a basis of The computation of is known to be an -hard problem For digraphs: Tutte’s 6-flow theorem Tutte’s 3-, 4-, 5-flow conjectures
Rank 3 (Points in the plane) Let be non-uniform (uniform case considered later) Theorem: Any connected co-simple non-uniform oriented matroid of rank 3 with more than 6 elements has trivial flow lattice (i. e. ). co-simple means (for rank 3): does not contain an -point line is the maximum regular oriented matroid of rank 3 The flow number is 2 A basis of is constructed inductively
The Uniform Case (Points in General Position) points do not share a hyperplane Any circuit has elements Example:
The Uniform Case (Points in General Position) For even rank (odd dimension) we have: (Hochstättler, Nešetřil 2003) Theorem: Let be a uniform oriented matroid of odd rank on elements. Then if and only if there is a reorientation ( ) so that There is a reorientation with balanced circuits: is a neighborly matroid polytope
The Uniform Case (Points in General Position) Theorem (structure of the lattice): Flow number: Basis construction: Let If is neighborly for all then is neighborly, too. Construct the basis inductively.
Summary Let be simple and co-simple on more than 6 elements
Outlook Does any (rank-preserving) single element extension of a (maximal) regular oriented matroid increase the dimension by ? What is the dimension of for general oriented matroids? Does always have a basis of signed circuits? Is there an orientable matroid so that but ? Otherwise would be well defined for orientable matroids.
Thanks for your attention