8 research outputs found
Matroid complexity and nonsuccinct descriptions
We investigate an approach to matroid complexity that involves describing a matroid via a list of independent sets, bases, circuits, or some other family of subsets of the ground set. The computational complexity of algorithmic problems under this scheme appears to be highly dependent on the choice of input type. We define an order on the various methods of description, and we show how this order acts upon 10 types of input. We also show that under this approach several natural algorithmic problems are complete in classes thought not to be equal to P
Matroids, Complexity and Computation
The node deletion problem on graphs is: given a graph and integer k, can we
delete no more than k vertices to obtain a graph that satisfies some property π.
Yannakakis showed that this problem is NP-complete for an infinite family of well-
defined properties. The edge deletion problem and matroid deletion problem are
similar problems where given a graph or matroid respectively, we are asked if we
can delete no more than k edges/elements to obtain a graph/matroid that satisfies
a property π. We show that these problems are NP-hard for similar well-defined
infinite families of properties.
In 1991 Vertigan showed that it is #P-complete to count the number of bases
of a representable matroid over any fixed field. However no publication has been
produced. We consider this problem and show that it is #P-complete to count
the number of bases of matroids representable over any infinite fixed field or finite
fields of a fixed characteristic.
There are many different ways of describing a matroid. Not all of these are
polynomially equivalent. That is, given one description of a matroid, we cannot
create another description for the same matroid in time polynomial in the size of
the first description. Due to this, the complexity of matroid problems can vary
greatly depending on the method of description used. Given one description a
problem might be in P while another description gives an NP-complete problem.
Based on these interactions between descriptions, we create and study the hierarchy
of all matroid descriptions and generalize this to all descriptions of countable
objects