2,046 research outputs found
One class of wild but brick-tame matrix problems
We present a class of wild matrix problems (representations of boxes), which
are "brick-tame," i.e. only have one-parameter families of \emph{bricks}
(representations with trivial endomorphism algebra). This class includes
several boxes that arise in study of simple vector bundles on degenerations of
elliptic curves, as well as those arising from the coadjoint action of some
linear groups.Comment: 19 page
Network Sparsification for Steiner Problems on Planar and Bounded-Genus Graphs
We propose polynomial-time algorithms that sparsify planar and bounded-genus
graphs while preserving optimal or near-optimal solutions to Steiner problems.
Our main contribution is a polynomial-time algorithm that, given an unweighted
graph embedded on a surface of genus and a designated face bounded
by a simple cycle of length , uncovers a set of size
polynomial in and that contains an optimal Steiner tree for any set of
terminals that is a subset of the vertices of .
We apply this general theorem to prove that: * given an unweighted graph
embedded on a surface of genus and a terminal set , one
can in polynomial time find a set that contains an optimal
Steiner tree for and that has size polynomial in and ; * an
analogous result holds for an optimal Steiner forest for a set of terminal
pairs; * given an unweighted planar graph and a terminal set , one can in polynomial time find a set that contains
an optimal (edge) multiway cut separating and that has size polynomial
in .
In the language of parameterized complexity, these results imply the first
polynomial kernels for Steiner Tree and Steiner Forest on planar and
bounded-genus graphs (parameterized by the size of the tree and forest,
respectively) and for (Edge) Multiway Cut on planar graphs (parameterized by
the size of the cutset). Additionally, we obtain a weighted variant of our main
contribution
On the diameter of random planar graphs
We show that the diameter D(G_n) of a random labelled connected planar graph
with n vertices is equal to n^{1/4+o(1)}, in probability. More precisely there
exists a constant c>0 such that the probability that D(G_n) lies in the
interval (n^{1/4-\epsilon},n^{1/4+\epsilon}) is greater than
1-\exp(-n^{c\epsilon}) for {\epsilon} small enough and n>n_0(\epsilon). We
prove similar statements for 2-connected and 3-connected planar graphs and
maps.Comment: 24 pages, 7 figure
Simple vector bundles on plane degenerations of an elliptic curve
In 1957 Atiyah classified simple and indecomposable vector bundles on an
elliptic curve. In this article we generalize his classification by describing
the simple vector bundles on all reduced plane cubic curves. Our main result
states that a simple vector bundle on such a curve is completely determined by
its rank, multidegree and determinant. Our approach, based on the
representation theory of boxes, also yields an explicit description of the
corresponding universal families of simple vector bundles
- …