A note on submodular set cover on matroids

We consider a problem related to the submodular set cover on polymatroids, when the ground set is the family of independent sets of a matroid. The achievement here is getting a strongly polynomial running time with respect to the ground set of the matroid even though the family of independent sets has exponential size. We also address the optimization problem of the maximization of submodular set functions on the independent sets of a matroid.     

The lattice of convex sets of an oriented matroid

Using the theory of the anti-exchange closure the structure of the lattice of convex sets of an acyclic oriented matroid is described. A new expression for the characteristic polynomial of the underlying matroid is proven using this analysis.     

Performance Bounds with Curvature for Batched Greedy Optimization

The batched greedy strategy is an approximation algorithm to maximize a set function subject to a matroid constraint. Starting with the empty set, the batched greedy strategy iteratively adds to the current solution set a batch of elements that results in the largest gain in the objective function while satisfying the matroid constraints. In this paper, we develop bounds on the performance of the batched greedy strategy relative to the optimal strategy in terms of a parameter called the total batched curvature. We show that when the objective function is a polymatroid set function, the batched greedy strategy satisfies a harmonic bound for a general matroid constraint and an exponential bound for a uniform matroid constraint, both in terms of the total batched curvature. We also study the behavior of the bounds as functions of the batch size. Specifically, we prove that the harmonic bound for a general matroid is nondecreasing in the batch size and the exponential bound for a uniform matroid is nondecreasing in the batch size under the condition that the batch size divides the rank of the uniform matroid. Finally, we illustrate our results by considering a task scheduling problem and an adaptive sensing problem.     

On Tutte polynomial expansion formulas in perspectives of matroids and oriented matroids

We introduce the active partition of the ground set of an oriented matroid perspective (or morphism, or quotient, or strong map) on a linearly ordered ground set. The reorientations obtained by arbitrarily reorienting parts of the active partition share the same active partition. This yields an equivalence relation for the set of reorientations of an oriented matroid perspective, whose classes are enumerated by coefficients of the Tutte polynomial, and a remarkable partition of the set of reorientations into Boolean lattices, from which we get a short direct proof of a 4-variable expansion formula for the Tutte polynomial in terms of orientation activities. This formula was given in the last unpublished preprint by Michel Las Vergnas; the above equivalence relation and notion of active partition generalize a former construction in oriented matroids by Michel Las Vergnas and the author; and the possibility of such a proof technique in perspectives was announced in the aforementioned preprint. We also briefly highlight how the 5-variable expansion of the Tutte polynomial in terms of subset activities in matroid perspectives comes in a similar way from the known partition of the power set of the ground set into Boolean lattices related to subset activities (and we complete the proof with a property which was missing in the literature). In particular, the paper applies to matroids and oriented matroids on a linearly ordered ground set, and applies to graph and directed graph homomorphisms on a linearly ordered edge-set.     

A rough set approach to the characterization of transversal matroids

Rough sets are efficient for data pre-processing during data mining. However, some important problems such as attribute reduction in rough sets are NP-hard and the algorithms required to solve them are mostly greedy ones. The transversal matroid is an important part of matroid theory, which provides well-established platforms for greedy algorithms. In this study, we investigate transversal matroids using the rough set approach. First, we construct a covering induced by a family of subsets and we propose the approximation operators and upper approximation number based on this covering. We present a sufficient condition under which a subset is a partial transversal, and also a necessary condition. Furthermore, we characterize the transversal matroid with the covering-based approximation operator and construct some types of circuits. Second, we explore the relationships between closure operators in transversal matroids and upper approximation operators based on the covering induced by a family of subsets. Finally, we study two types of axiomatic characterizations of the covering approximation operators based on the set theory and matroid theory, respectively. These results provide more methods for investigating the combination of transversal matroids with rough sets.     

Polynomial Time Recognition of Uniform Cocircuit Graphs

We present an algorithm which takes a graph as input and decides in polynomial time if the graph is the cocircuit graph of a uniform oriented matroid. In the affirmative case the algorithm returns the set of signed cocircuits of the oriented matroid.     

A Linear Programming Construction of Fully Optimal Bases in Graphs and Hyperplane Arrangements

The fully optimal basis of a bounded acyclic oriented matroid on a linearly ordered set has been defined and studied by the present authors in a series of papers, dealing with graphs, hyperplane arrangements, and oriented matroids (in order of increasing generality). This notion provides a bijection between bipolar orientations and uniactive internal spanning trees in a graph resp. bounded regions and uniactive internal bases in a hyperplane arrangement or an oriented matroid (in the sense of Tutte activities). This bijection is the basic case of a general activity preserving bijection between reorientations and subsets of an oriented matroid, called the active bijection, providing bijective versions of various classical enumerative results.Fully optimal bases can be considered as a strenghtening of optimal bases from linear programming, with a simple combinatorial definition. Our first construction used this purely combinatorial characterization, providing directly an algorithm to compute in fact the reverse bijection. A new definition uses a direct construction in terms of a linear programming. The fully optimal basis optimizes a sequence of nested faces with respect to a sequence of objective functions (whereas an optimal basis in the usual sense optimizes one vertex with respect to one objective function). This note presents this construction in terms of graphs and linear algebra.     

On delaunay oriented matroids for convex distance functions

For any finite point setS inE ^d, an oriented matroid DOM (S) can be defined in terms of howS is partitioned by Euclidean hyperspheres. This oriented matroid is related to the Delaunay triangulation ofS and is realizable, because of thelifting property of Delaunay triangulations. We prove that the same construction of aDelaunay oriented matroid can be performed with respect to any smooth, strictly convex distance function in the planeE ² (Theorem 3.5). For these distances, the existence of a Delaunay oriented matroid cannot follow from a lifting property, because Delaunay triangulations might be nonregular (Theorem 4.2(i). This is related to the fact that the Delaunay oriented matroid can be nonrealizable (Theorem 4.2(ii). This research was partially supported by the Spanish Grant DGICyT PB 92/0498-C02 and the David and Lucile Packard Foundation.     

Checking oriented matroid isomorphism by means of canonical labeling

In this paper a method for establishing the structural equivalence of sets of planar geometric features composed by points and lines is presented. It is based on oriented matroid theory, a setting in which the combinatorial structural properties of these geometric features, such as incidence, order, partitioning, separation, and convexity, can be represented and analyzed in a coordinate-free manner. Projective transformations in computer vision keep in general the convexity property which implies an invariant oriented matroid representation of the planar geometric features under this class of transformations. As long as points and lines are in general position, the oriented matroid representation is also insensitive to small changes in the geometric image features. However the oriented matroid representation depends on the labeling of its elements. Checking the structural equivalence of the above mentioned geometric features represented by means of oriented matroids implies establishing whether two oriented matroid representations are equivalent up to relabeling of their elements. This is the oriented matroid isomorphism problem which is solved in this paper by means of a canonical labeling of the elements.     

The switching game on unions of oriented matroids

In 1986, Hamidoune and Las Vergnas [3] introduced an oriented matroid version of the so-called Shannon’s switching game. They conjectured that the classification of the directed switching game on oriented matroids is identical to the classification of the non-oriented version. In this note, we support this conjecture by showing its validity for an infinite class of oriented matroids obtained as unions of rank-1 and/or rank-2 uniform oriented matroids.     

Greedy Families for Linear Objective Functions

The following structures are characterized: for which families of feasible subsets of a finite set does the greedy algorithm return the optimum subset independent of the weighting of a linear objective function on the set? Characteristically, the family must then have as bases the bases of a matroid (even when the feasible family is not a system of independent sets), and for every accessible feasible set X, the subset of elements by which X can be augmented is the complement of a proper closed set of the matroid. Another characterization is given for a family in which the greedy algorithm gives the optimum subset at every stage: the family is that of the bases of a sequence of matroid strong maps resulting in a natural duality theory. Theoretical underpinnings are given for several classical instances such as the algorithms of Kruskal, Prim, and Dijkstra.     

Orientability of matroids

In this paper we define oriented matroids and develop their fundamental properties, which lead to generalizations of known results concerning directed graphs, convex polytopes, and linear programming. Duals and minors of oriented matroids are defined. It is shown that every coordinatization (representation) of a matroid over an ordered field induces an orientation of the matroid. Examples of matroids that are orientable but not coordinatizable and of matroids that are not orientable are presented. We show that a binary matroid is orientable if and only if it is unimodular (regular), and that every unimodular matroid has an orientation that is induced by a coordinatization and is unique in a certain straightforward sense.     

Oriented matroids, complex manifolds, and a combinatorial model for BU

We introduce a new notion of complex oriented matroid and develop some basic properties of this object. Our definition of complex oriented matroids bears the same relationship to classical oriented matroids that the stratification of the complex plane into nine components corresponding to the signs of the complex and real parts has with the three-component sign stratification of the real line. We then use these complex oriented matroids to set up the foundations of a combinatorial version of complex geometry analogous to MacPherson's combinatorial differential manifolds; in this world, the representing object for the functor of (combinatorial) complex vector bundles is the nerve of a poset of complex oriented matroids. We conclude by showing that this space is homotopy equivalent to the complex Grassmannian, thus deducing that our combinatorial world is able to completely capture the notion of complex vector bundles.     

Oriented matroids and multiply ordered sets

The many different axiomatizations for matroids all have their uses. In this paper we show that Gutierrez Novoa's n-ordered sets are cryptomorphically the same as the oriented matroids, thereby establishing the existence of an axiomatization for oriented matroids in which the “oriented” bases of the matroid are the objects of paramount importance.     

Sample path large deviations and intree networks

Using the contraction principle, in this paper we derive a set of closure properties for sample path large deviations. These properties include sum, reduction, composition and reflection mapping. Using these properties, we show that the exponential decay rates of the steady state queue length distributions in an intree network with routing can be derived by a set of recursive equations. The solution of this set of equations is related to the recently developed theory of effective bandwidth for high speed digital networks, especially ATM networks. We also prove a conditional limit theorem that illustrates how a queue builds up in an intree network.     

Euler's relation,möbius functions and matroid identities

We give a short combinatorial proof of the Euler relation for convex polytopes in the context of oriented matroids. Using counting arguments we derive from the Euler relation several identities holding in the lattice of flats of an oriented matroid. These identities are proven for any matroid by Möbius inversion.     

Fully Optimal Bases and the Active Bijection in Graphs,Hyperplane Arrangements,and Oriented Matroids

In this note, we present the main results of a series of forthcoming papers, dealing with bi-jective 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 bijec-tion 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.     

A GENERAL DETERMINISTIC PIVOT METHOD FOR ORIENTED MATROID PROGRAMMING

A new pivot method for oriented matroid progiamming is given out. This mathod is deterministic by nature and is general in the sense that its flexible pivot selection rule allows a family of possible algorithms to be its special cases, including the so called criss-cross algorithm and the Edmonds-Fukuda algorithm as well. As an example of a special implementation of our general method, an extended version of the Edmonds-Fukuda algorithm is presented.     

Matroid Polytopes and their Volumes

We express the matroid polytope P _M of a matroid M as a signed Minkowski sum of simplices, and obtain a formula for the volume of P _M . This gives a combinatorial expression for the degree of an arbitrary torus orbit closure in the Grassmannian Gr _k,n . We then derive analogous results for the independent set polytope and the underlying flag matroid polytope of M. Our proofs are based on a natural extension of Postnikov’s theory of generalized permutohedra.     

Oriented matroids and Ky Fan’s theorem

L. Lovász (Matroids and Sperner’s Lemma, Europ. J. Comb. 1 (1980), 65–66) has shown that Sperner’s combinatorial lemma admits a generalization involving a matroid defined on the set of vertices of the associated triangulation. We prove that Ky Fan’s theorem admits an oriented matroid generalization of similar nature. Classical Ky Fan’s theorem is obtained as a corollary if the underlying oriented matroid is chosen to be the alternating matroid C ^m,r .