Matroid Research Papers - Academia.edu (original) (raw)

We study some properties of a serial (i.e. one-by-one) symmetric exchange of elements of two disjoint bases of a matroid. We show that any two elements of one base have a serial symmetric exchange with some two elements of the other base.... more

We study some properties of a serial (i.e. one-by-one) symmetric exchange of elements of two disjoint bases of a matroid. We show that any two elements of one base have a serial symmetric exchange with some two elements of the other base. As a result, we obtain that any two disjoint bases in a matroid of rank 4 have a full serial symmetric exchange.

Abstract. From 22.08.04 to 27.08.04, the Dagstuhl Seminar 04351 Spa-

We consider 'source location problems' in undirected graphs motivated by localization problems in sensor networks. In such a network the fundamental problem is to determine the locations of the sensors in the plane from a subset... more

We consider 'source location problems' in undirected graphs motivated by localization problems in sensor networks. In such a network the fundamental problem is to determine the locations of the sensors in the plane from a subset of pairwise distances. To achieve unique localizability it is necessary to designate a set of sensors, called anchors, for which the exact location is known. We consider the problem of finding a smallest set of anchors which make the network uniquely localizable, provided that the coordinates are 'generic'. We give polynomial time algorithms for two relaxations of the problem. By combining these algorithms we obtain a 2-approximation algorithm for the anchor minimization problem.

Abstract: We consider ‘source location problems ’ in undirected graphs motivated by localization problems in sensor networks. In such a network the fundamental problem is to determine the locations of the sensors in the plane from a... more

Abstract: We consider ‘source location problems ’ in undirected graphs motivated by localization problems in sensor networks. In such a network the fundamental problem is to determine the locations of the sensors in the plane from a subset of pairwise distances. To achieve unique localizability it is necessary to designate a set of sensors, called anchors, for which the exact location is known. We consider the problem of finding a smallest set of anchors which make the network uniquely localizable, provided that the coordinates are ‘generic’. We give polynomial time algorithms for two relaxations of the problem. By combining these algorithms we obtain a 2-approximation algorithm for the anchor minimization problem.

We present a number of combinatorial characterizations of K-matrices. This extends a theorem of Fiedler and Ptak on linear-algebraic characterizations of K-matrices to the setting of oriented matroids. Our proof is elementary and... more

We present a number of combinatorial characterizations of K-matrices. This extends a theorem of Fiedler and Ptak on linear-algebraic characterizations of K-matrices to the setting of oriented matroids. Our proof is elementary and simplifies the original proof substantially by exploiting the duality of oriented matroids. As an application, we show that a simple principal pivot method applied to the linear complementarity problems with K-matrices converges very quickly, by a purely combinatorial argument.

We present generalisations of several MacWilliams type identities, including those by Klve and Shiromoto, and of the theorems of Greene and Barg that describe support weight enumerators of the code. One of our main tools is a... more

We present generalisations of several MacWilliams type identities, including those by Klve and Shiromoto, and of the theorems of Greene and Barg that describe support weight enumerators of the code. One of our main tools is a generalisation of a decomposition theorem due to Brylawski. 1

We present a randomized linear-time algorithm to find a minimum spanning tree in a connected graph with edge weights. The algorithm uses random sampling in combination with a recently discovered linear-time algorithm for verifying a... more

We present a randomized linear-time algorithm to find a minimum spanning tree in a connected graph with edge weights. The algorithm uses random sampling in combination with a recently discovered linear-time algorithm for verifying a minimum spanning tree. Our computational model is a unit-cost random-access machine with the restriction that the only operations allowed on edge weights are binary comparisons.

Oriented matroids are combinatorial structures that generalize point configurations, vector configurations, hyperplane arrangements, polyhedra, linear programs, and directed graphs. Oriented matroids have played a key role in... more

Oriented matroids are combinatorial structures that generalize point configurations, vector configurations, hyperplane arrangements, polyhedra, linear programs, and directed graphs. Oriented matroids have played a key role in combinatorics, computational geometry, and optimization. This paper surveys prior work and presents an update on the search for bounds on the diameter of the cocircuit graph of an oriented matroid. The motivation for our investigations is the complexity of the simplex method and the criss-cross method. We review the diameter problem and show the diameter bounds of general oriented matroids reduce to those of uniform oriented matroids. We give the latest exact bounds for oriented matroids of low rank and low corank, and for all oriented matroids with up to nine elements (this part required a large computer-based proof). For arbitrary oriented matroids, we present an improvement to a quadratic bound of Finschi. Our discussion highlights an old conjecture that s...