We consider the number of queries needed to solve the matroid intersection problem, a question raised by Welsh (1976). Given two matroids of rank r on n elements, it is known that O(nr) independence queries suffice. Unfortunately, very little is known about lower bounds for this problem. This paper describes three lower bounds which, to our knowledge, are the best known: 2n− 2 queries are neede...

We prove that the mathematical framework for the de Sitter top system is the de Sitter fiber bundle. In this context, the concept of soldering associated with a fiber bundle plays a central role. We comment on the possibility that our formalism may be of particular interest in different contexts including MacDowell-Mansouri theory, two time physics and oriented matroid theory.

We survey some lower bounds on the function in the title based on matroid theory and address the following problem by Dosa, Szalkai, Laflamme [9]: Determine the smallest number of circuits in a loopless matroid with no parallel elements and with a given size and rank. In the graphic 3-connected case we provide a lower bound which is a product of a linear function of the number of vertices and a...

In a 1965 paper, Erdős remarked that a graph G has a bipartite subgraph that has at least half the number of edges of G. The purpose of this note is to prove a matroid analogue of Erdős’s original observation. It follows from this matroid result that every loopless binary matroid has a restriction that uses more than half of its elements and has no odd circuits; and, for 2 ≤ k ≤ 5, every bridge...

In this paper, we show that for any independence system, the problem of finding a persistency partition of the ground set and that of finding a maximum weight independent set are polynomially equivalent.

Combinatorial Optimization is a central sub-area in Operations Research that has found many applications in computational biology. In this talk I will survey some of my research in computational biology that uses graph theory, matroid theory, and integer linear programming. The biological applications come from haplotyping, the study of recombination and recombination networks, and phylogenetic...

We analyze maximal supersymmetry in eleven-dimensional supergravity from the point of view of the oriented matroid theory. The mathematical key tools in our discussion are the Englert solution and the chirotope concept. We argue that chirotopes may provide other solutions not only for elevendimensional supergravity but for any higher dimensional supergravity theory.

In 1991, Wei introduced generalized minimum Hamming weights for linear codes and showed their monotonicity and duality. Recently, several authors extended these results to the case of generalized minimum poset weights by using different methods. Here, we would like to prove the duality by using matroid theory. This gives yet another and very simple proof of it. In particular, our argument will ...

We construct an intersection product on tropical cycles contained in the Bergman fan of a matroid. To do this we first establish a connection between the operations of deletion and restriction in matroid theory and tropical modifications as defined by Mikhalkin in [14]. This product generalises the product of Allermann and Rau [2], and Allermann [1] and also provides an alternative procedure fo...

Recently, locally repairable codes has gained significant interest for their potential applications in distributed storage systems. However, most constructions in existence are over fields with size that grows with the number of servers, which makes the systems computationally expensive and difficult to maintain. Here, we study linear locally repairable codes over the binary field, tolerating m...