35 research outputs found
Blocking optimal -arborescences
Given a digraph and a positive integer , an arc set is called a \textbf{-arborescence} if it is the disjoint union of
spanning arborescences. The problem of finding a minimum cost -arborescence
is known to be polynomial-time solvable using matroid intersection. In this
paper we study the following problem: find a minimum cardinality subset of arcs
that contains at least one arc from every minimum cost -arborescence. For
, the problem was solved in [A. Bern\'ath, G. Pap , Blocking optimal
arborescences, IPCO 2013]. In this paper we give an algorithm for general
that has polynomial running time if is fixed
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)
Covering symmetric skew-supermodular functions with hyperedges
In this paper we give results related to a theorem of Szigeti that concerns
the covering of symmetric skew-supermodular set functions with hyperedges of
minimum total size. In particular, we show the following generalization using a
variation of Schrijver’s supermodular colouring theorem: if p1 and p2 are skewsupermodular
functions whose maximum value is the same, then it is possible to
find in polynomial time a hypergraph of minimum total size that covers both of
them. Note that without the assumption on the maximum values this problem
is NP-hard. The result has applications concerning the local edge-connectivity
augmentation problem of hypergraphs and the global edge-connectivity augmentation
problem of mixed hypergraphs. We also present some results on the case
when the hypergraph must be obtained by merging given hyperedges