176 research outputs found
Matroid packing and covering with circuits through an element
In 1981, Seymour proved a conjecture of Welsh that, in a connected matroid M, the sum of the maximum number of disjoint circuits and the minimum number of circuits needed to cover M is at most r*(M) + 1. This paper considers the set Ce(M) of circuits through a fixed element e such that M/e is connected. Let νe(M) be the maximum size of a subset of Ce(M) in which any two distinct members meet only in {e}, and let θe(M) be the minimum size of a subset of Ce(M) that covers M. The main result proves that νe(M) + θe(M) ≤ r* + 2 and that if M has no Fano-minor using e, then νe + θe,(M) ≤ r*(M) + 1. Seymour\u27s result follows without difficulty from this theorem and there are also some interesting applications to graphs. © 2005 Elsevier Inc. All rights reserved
Even an infinite bureaucracy eventually makes a decision
We show that the fact that a political decision filtered through a finite
tree of committees gives a determined answer generalises in some sense to
infinite trees. This implies a new special case of the Matroid Intersection
Conjecture
On the tractability of some natural packing, covering and partitioning problems
In this paper we fix 7 types of undirected graphs: paths, paths with
prescribed endvertices, circuits, forests, spanning trees, (not necessarily
spanning) trees and cuts. Given an undirected graph and two "object
types" and chosen from the alternatives above, we
consider the following questions. \textbf{Packing problem:} can we find an
object of type and one of type in the edge set of
, so that they are edge-disjoint? \textbf{Partitioning problem:} can we
partition into an object of type and one of type ?
\textbf{Covering problem:} can we cover with an object of type
, and an object of type ? This framework includes 44
natural graph theoretic questions. Some of these problems were well-known
before, for example covering the edge-set of a graph with two spanning trees,
or finding an - path and an - path that are
edge-disjoint. However, many others were not, for example can we find an
- path and a spanning tree that are
edge-disjoint? Most of these previously unknown problems turned out to be
NP-complete, many of them even in planar graphs. This paper determines the
status of these 44 problems. For the NP-complete problems we also investigate
the planar version, for the polynomial problems we consider the matroidal
generalization (wherever this makes sense)
On the Matroid Intersection Conjecture
In this dissertation, we investigate the Matroid Intersection Conjecture for pairs of matroids on the same ground set, proposed by Nash-Williams in 1990. Originally, the conjecture was stated for finitary matroids only, but we consider it for general matroids and introduce new approaches to attack the conjecture.;The first approach is to consider the situation when it is possible to make a finite modification to the matroids after which the pair satisfies the conjecture. In such a situation we say that the pair has the Almost Intersection Property. We prove that any pair of matroids with the Almost Intersection Property must satisfy the Matroid Intersection Conjecture. Using this result we prove that the Matroid Intersection Conjecture is true in the case when one of the matroids has finite rank and also in the case when one of the matroids is a patchwork matroid.;Our second new approach is inspired by the proof of the general version of Konig\u27s Theorem for bipartite graphs. That result implies that the Matroid Intersection Conjecture is true for pairs of partition matroids. We develop some new techniques that generalize the critical set approach used in the proof of the countable version of Konig\u27s Theorem. Our results enable us to prove that the Matroid Intersection Conjecture is true for a pair of singular matroids on a set that is infinitely countable. A matroid is singular when it is a direct sum of matroids such that each term of the sum is a uniform matroid either of rank one or of co-rank one
Preprocessing under uncertainty
In this work we study preprocessing for tractable problems when part of the
input is unknown or uncertain. This comes up naturally if, e.g., the load of
some machines or the congestion of some roads is not known far enough in
advance, or if we have to regularly solve a problem over instances that are
largely similar, e.g., daily airport scheduling with few charter flights.
Unlike robust optimization, which also studies settings like this, our goal
lies not in computing solutions that are (approximately) good for every
instantiation. Rather, we seek to preprocess the known parts of the input, to
speed up finding an optimal solution once the missing data is known.
We present efficient algorithms that given an instance with partially
uncertain input generate an instance of size polynomial in the amount of
uncertain data that is equivalent for every instantiation of the unknown part.
Concretely, we obtain such algorithms for Minimum Spanning Tree, Minimum Weight
Matroid Basis, and Maximum Cardinality Bipartite Maxing, where respectively the
weight of edges, weight of elements, and the availability of vertices is
unknown for part of the input. Furthermore, we show that there are tractable
problems, such as Small Connected Vertex Cover, for which one cannot hope to
obtain similar results.Comment: 18 page
Matroid and Tutte-connectivity in infinite graphs
We relate matroid connectivity to Tutte-connectivity in an infinite graph.
Moreover, we show that the two cycle matroids, the finite-cycle matroid and the
cycle matroid, in which also infinite cycles are taken into account, have the
same connectivity function. As an application we re-prove that, also for
infinite graphs, Tutte-connectivity is invariant under taking dual graphs.Comment: 11 page
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