https://hal-lirmm.ccsd.cnrs.fr/lirmm-00154521Gioan, EmericEmericGioanALGCO - Algorithmes, Graphes et Combinatoire - LIRMM - Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier - UM - Université de Montpellier - CNRS - Centre National de la Recherche ScientifiqueLas Vergnas, MichelMichelLas VergnasC&O - Equipe combinatoire et optimisation - IMJ-PRG - Institut de Mathématiques de Jussieu - Paris Rive Gauche - UPMC - Université Pierre et Marie Curie - Paris 6 - UPD7 - Université Paris Diderot - Paris 7 - CNRS - Centre National de la Recherche ScientifiqueFully optimal bases and the active bijection in graphs, hyperplane arrangements, and oriented matroidsHAL CCSD2007oriented matroidhyperplane arrangementgraphTutte polynomiallinear programmingoptimal basisbijectionreorientationbasisregionno broken circuit[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM][MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO]Gioan, Emeric2007-06-13 23:55:442022-09-06 17:00:302007-06-14 12:48:45enConference papers1In this note, we present the main results of a series of forthcoming papers, dealing with bijective generalizations of some counting formulas. New intrinsic constructions in oriented matroids on a linearly ordered set of elements establish notably structural links between counting regions and linear programming. We introduce 'fully optimal bases', which have a simple combinatorial characterization, and strengthen the well-known optimal bases of linear programming. Our main result is that every bounded region of an ordered hyperplane arrangement, or ordered oriented matroid, has a unique fully optimal basis, providing the 'active bijection' between bounded regions and uniactive internal bases. The active bijection is extended to an activity preserving mapping between all reorientations and all bases of an ordered oriented matroid. It gives a bijective interpretation of the equality of two expressions for the Tutte polynomial, as well as a new expression of this polynomial in terms of beta invariants of minors. There are several refinements, such as an activity preserving bijection between regions (acyclic reorientations) and no-broken-circuit subsets, and others in terms of hyperplane arrangements, graphs, and permutations.