Latroids and their representation by codes over modules

It has been known for some time that there is a connection between linear codes over fields and matroids represented over fields. In fact a generator matrix for a linear code over a field is also a representation of a matroid over that field. There are intimately related operations of deletion, contraction, minors and duality on both the code and the matroid. The weight enumerator of the code is an evaluation of the Tutte polynomial of the matroid, and a standard identity relating the Tutte polynomials of dual matroids gives rise to a MacWilliams identity relating the weight enumerators of dual codes. More recently, codes over rings and modules have been considered, and MacWilliams type identities have been found in certain cases.

In this paper we consider codes over rings and modules with code duality based on a Morita duality of categories of modules. To these we associate latroids, defined here. We generalize notions of deletion, contraction, minors and duality, on both codes and latroids, and examine all natural relations among these.

We define generating functions associated with codes and latroids, and prove identities relating them, generalizing above-mentioned generating functions and identities.

     

Coordinatewise domain scaling algorithm for M-convex function minimization

We present a polynomial time scaling algorithm for the minimization of an M-convex function. M-convex functions are nonlinear discrete functions with (poly)matroid structures, which are being recognized as playing a fundamental role in tractable cases of discrete optimization. The algorithm is applicable also to a variant of quasi M-convex functions.Mathematics Subject Classification (2000):90C27, 68W40, 05B35This work is supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan.     

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.     

On the intersection of infinite matroids

We show that the infinite matroid intersection conjecture of Nash-Williams implies the infinite Menger theorem proved by Aharoni and Berger in 2009.We prove that this conjecture is true whenever one matroid is nearly finitary and the second is the dual of a nearly finitary matroid, where the nearly finitary matroids form a superclass of the finitary matroids.In particular, this proves the infinite matroid intersection conjecture for finite-cycle matroids of 2-connected, locally finite graphs with only a finite number of vertex-disjoint rays.     

A matroid-friendly basis for the quasisymmetric functions

A new Z-basis for the space of quasisymmetric functions (QSym, for short) is presented. It is shown to have nonnegative structure constants, and several interesting properties relative to the quasisymmetric functions associated to matroids by the Hopf algebra morphism F of Billera, Jia, and Reiner [L.J. Billera, N. Jia, V. Reiner, A quasisymmetric function for matroids, arXiv:math.CO/0606646]. In particular, for loopless matroids, this basis reflects the grading by matroid rank, as well as by the size of the ground set. It is shown that the morphism F distinguishes isomorphism classes of rank two matroids, and that decomposability of the quasisymmetric function of a rank two matroid mirrors the decomposability of its base polytope. An affirmative answer to the Hilbert basis question raised in [L.J. Billera, N. Jia, V. Reiner, A quasisymmetric function for matroids, arXiv:math.CO/0606646] is given.     

A simple approach to some basic results in matroid theory

The bases and the cocircuits of a matroid form a blocking pair of clutters; this fact leads to simple proofs of some basic and well-known facts about matroids, including a variety of axiomatizations.     

Notes on L-/M-convex functions and the separation theorems

The concepts of L-convex function and M-convex function have recently been introduced by Murota as generalizations of submodular function and base polyhedron, respectively, and discrete separation theorems are established for L-convex/concave functions and for M-convex/concave functions as generalizations of Frank’s discrete separation theorem for submodular/supermodular set functions and Edmonds’ matroid intersection theorem. This paper shows the equivalence between Murota’s L-convex functions and Favati and Tardella’s submodular integrally convex functions, and also gives alternative proofs of the separation theorems that provide a geometric insight by relating them to the ordinary separation theorem in convex analysis. Received: November 27, 1997 / Accepted: December 16, 1999?Published online May 12, 2000     

Persistency in combinatorial optimization problems on matroids

Optimization problems on matroids are generalizations of such important combinatorial optimization problems like the problem of minimum spanning tree of a graph, the bipartite matching problem, flow problems, etc. We analyze algorithms for finding the maximum weight independent set of a matroid and for finding a maximum cardinality intersection of two matroids and extend them to obtain the so-called “persistency” partition of the basic set of the matroid, where contain elements belonging to all optimum solutions; contain elements not belonging to any optimum solution; contain elements that belong to some but not to all optimum solutions.     

Matroid matching in pseudomodular lattices

The matroid matching problem (also known as matroid parity problem) has been intensively studied by several authors. Starting from very special problems, in particular the matching problem and the matroid intersection problem, good characterizations have been obtained for more and more general classes of matroids. The two most recent ones are the class of representable matroids and, later on, the class of algebraic matroids (cf. [4] and [2]). We present a further step of generalization, showing that a good characterization can also be obtained for the class of socalled pseudomodular matroids, introduced by Björner and Lovász (cf. [1]). A small counterexample is included to show that pseudomodularity still does not cover all matroids that behave well with respect to matroid matching.Supported by the German Research Association (Deutsche Forschungsgemeinschaft, SFB 303).     

Four matroidal structures of covering and their relationships with rough sets

Covering is a common form of data representation, and covering-based rough sets serve as an efficient technique to process this type of data. However, many important problems such as covering reduction in covering-based rough sets are NP-hard so that most algorithms to solve them are greedy. Matroids provide well-established platforms for greedy algorithm foundation and implementation. Therefore, it is necessary to integrate covering-based rough set with matroid. In this paper, we propose four matroidal structures of coverings and establish their relationships with rough sets. First, four different viewpoints are presented to construct these four matroidal structures of coverings, including 1-rank matroids, bigraphs, upper approximation numbers and transversals. The respective advantages of these four matroidal structures to rough sets are explored. Second, the connections among these four matroidal structures are studied. It is interesting to find that they coincide with each other. Third, a converse view is provided to induce a covering by a matroid. We study the relationship between this induction and the one from a covering to a matroid. Finally, some important concepts of covering-based rough sets, such as approximation operators, are equivalently formulated by these matroidal structures. These interesting results demonstrate the potential to combine covering-based rough sets with matroids.     

Topological representations of matroid maps

The Topological Representation Theorem for (oriented) matroids states that every (oriented) matroid arises from the intersection lattice of an arrangement of codimension one homotopy spheres on a homotopy sphere. In this paper, we use a construction of Engström to show that structure-preserving maps between matroids induce topological mappings between their representations; a result previously known only in the oriented case. Specifically, we show that weak maps induce continuous maps and that this process is a functor from the category of matroids with weak maps to the homotopy category of topological spaces. We also give a new and conceptual proof of a result regarding the Whitney numbers of the first kind of a matroid.     

Matroids and linking systems

With the help of the concept of a linking system, theorems relating matroids with bipartite graphs and directed graphs are deduced. In this way natural generalizations of theorems of Edmonds & Fulkerson, Perfect, Pym. Rado, Brualdi and Mason are obtained. Furthermore some other properties of these linking systems are investigated.     

The Lattice of Cyclic Flats of a Matroid

A flat of a matroid is cyclic if it is a union of circuits. The cyclic flats of a matroid form a lattice under inclusion. We study these lattices and explore matroids from the perspective of cyclic flats. In particular, we show that every lattice is isomorphic to the lattice of cyclic flats of a matroid. We give a necessary and sufficient condition for a lattice of sets and a function to be the lattice of cyclic flats of a matroid and the restriction of the corresponding rank function to . We apply this perspective to give an alternative view of the free product of matroids and we show how to compute the Tutte polynomial of the free product in terms of the Tutte polynomials of the constituent matroids. We define cyclic width and show that this concept gives rise to minor-closed, dual-closed classes of matroids, two of which contain only transversal matroids. Received May 29, 2005     

A characterization of orthogonal duality in matroid theory

An operation on matroids is a function defined from the collection of all matroids on finite sets to itself which preserves isomorphism of matroids and sends a matroid on a set S to a matroid on the same set S. We show that orthogonal duality is the only non-trivial operation on matroids which interchanges contraction and deletion.     

Oriented matroid systems

The domination invariant has played an important part in reliability theory. While most of the work in this field has been restricted to various types of network system models, many of the results can be generalized to much wider families of systems associated with matroids. Previous papers have explored the relation between undirected network systems and matroids. In this paper the main focus is on directed network systems and their relation to oriented matroids. An oriented matroid is a special type of matroid where the circuits are signed sets. Using these signed sets one can e.g., obtain a set theoretic representation of the direction of the edges of a directed network system. Classical results for directed network systems include the fact that the signed domination is either +1 or −1 if the network is acyclic, and zero otherwise. It turns out that these results can be generalized to systems derived from oriented matroids. Several classes of systems for which the generalized results hold will be discussed. These include oriented versions of k-out-of-n systems and a certain class of systems associated with matrices.     

Fast scaling algorithms for M-convex function minimization with application to the resource allocation problem

M-convex functions, introduced by Murota (Adv. Math. 124 (1996) 272; Math. Prog. 83 (1998) 313), enjoy various desirable properties as “discrete convex functions.” In this paper, we propose two new polynomial-time scaling algorithms for the minimization of an M-convex function. Both algorithms apply a scaling technique to a greedy algorithm for M-convex function minimization, and run as fast as the previous minimization algorithms. We also specialize our scaling algorithms for the resource allocation problem which is a special case of M-convex function minimization.     

Quasi‐graphic matroids*

Frame matroids and lifted‐graphic matroids are two interesting generalizations of graphic matroids. Here, we introduce a new generalization, quasi‐graphic matroids, that unifies these two existing classes. Unlike frame matroids and lifted‐graphic matroids, it is easy to certify that a 3‐connected matroid is quasi‐graphic. The main result is that every 3‐connected representable quasi‐graphic matroid is either a lifted‐graphic matroid or a frame 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.