8,844 research outputs found
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)
Some NP-complete edge packing and partitioning problems in planar graphs
Graph packing and partitioning problems have been studied in many contexts,
including from the algorithmic complexity perspective. Consider the packing
problem of determining whether a graph contains a spanning tree and a cycle
that do not share edges. Bern\'ath and Kir\'aly proved that this decision
problem is NP-complete and asked if the same result holds when restricting to
planar graphs. Similarly, they showed that the packing problem with a spanning
tree and a path between two distinguished vertices is NP-complete. They also
established the NP-completeness of the partitioning problem of determining
whether the edge set of a graph can be partitioned into a spanning tree and a
(not-necessarily spanning) tree. We prove that all three problems remain
NP-complete even when restricted to planar graphs.Comment: 6 pages, 2 figure
Vertex Disjoint Path in Upward Planar Graphs
The -vertex disjoint paths problem is one of the most studied problems in
algorithmic graph theory. In 1994, Schrijver proved that the problem can be
solved in polynomial time for every fixed when restricted to the class of
planar digraphs and it was a long standing open question whether it is
fixed-parameter tractable (with respect to parameter ) on this restricted
class. Only recently, \cite{CMPP}.\ achieved a major breakthrough and answered
the question positively. Despite the importance of this result (and the
brilliance of their proof), it is of rather theoretical importance. Their proof
technique is both technically extremely involved and also has at least double
exponential parameter dependence. Thus, it seems unrealistic that the algorithm
could actually be implemented. In this paper, therefore, we study a smaller
class of planar digraphs, the class of upward planar digraphs, a well studied
class of planar graphs which can be drawn in a plane such that all edges are
drawn upwards. We show that on the class of upward planar digraphs the problem
(i) remains NP-complete and (ii) the problem is fixed-parameter tractable.
While membership in FPT follows immediately from \cite{CMPP}'s general result,
our algorithm has only single exponential parameter dependency compared to the
double exponential parameter dependence for general planar digraphs.
Furthermore, our algorithm can easily be implemented, in contrast to the
algorithm in \cite{CMPP}.Comment: 14 page
Arc-Disjoint Paths and Trees in 2-Regular Digraphs
An out-(in-)branching B_s^+ (B_s^-) rooted at s in a digraph D is a connected
spanning subdigraph of D in which every vertex x != s has precisely one arc
entering (leaving) it and s has no arcs entering (leaving) it. We settle the
complexity of the following two problems:
1) Given a 2-regular digraph , decide if it contains two arc-disjoint
branchings B^+_u, B^-_v.
2) Given a 2-regular digraph D, decide if it contains an out-branching B^+_u
such that D remains connected after removing the arcs of B^+_u.
Both problems are NP-complete for general digraphs. We prove that the first
problem remains NP-complete for 2-regular digraphs, whereas the second problem
turns out to be polynomial when we do not prescribe the root in advance. We
also prove that, for 2-regular digraphs, the latter problem is in fact
equivalent to deciding if contains two arc-disjoint out-branchings. We
generalize this result to k-regular digraphs where we want to find a number of
pairwise arc-disjoint spanning trees and out-branchings such that there are k
in total, again without prescribing any roots.Comment: 9 pages, 7 figure
Label-connected graphs and the gossip problem
A graph with m edges is called label-connected if the edges can be labeled with real numbers in such a way that, for every pair (u, v) of vertices, there is a (u, v)-path with ascending labels. The minimum number of edges of a label-connected graph on n vertices equals the minimum number of calls in the gossip problem for n persons, which is known to be 2n − 4 for n ≥ 4. A polynomial characterization of label-connected graphs with n vertices and 2n − 4 edges is obtained. For a graph G, let θ(G) denote the minimum number of edges that have to be added to E(G) in order to create a graph with two edge-disjoint spanning trees. It is shown that for a graph G to be label-connected, θ(G) ≤ 2 is necessary and θ(G) ≤ 1 is sufficient. For i = 1, 2, the condition θ(G) ≤ i can be checked in polynomial time. Yet recognizing label-connected graphs is an NP-complete problem. This is established by first showing that the following problem is NP-complete: Given a graph G and two vertices u and v of G, does there exist a (u, v)-path P in G such that G−E(P) is connected
The maximum disjoint paths problem on multi-relations social networks
Motivated by applications to social network analysis (SNA), we study the
problem of finding the maximum number of disjoint uni-color paths in an
edge-colored graph. We show the NP-hardness and the approximability of the
problem, and both approximation and exact algorithms are proposed. Since short
paths are much more significant in SNA, we also study the length-bounded
version of the problem, in which the lengths of paths are required to be upper
bounded by a fixed integer . It is shown that the problem can be solved in
polynomial time for and is NP-hard for . We also show that the
problem can be approximated with ratio in polynomial time
for any . Particularly, for , we develop an efficient
2-approximation algorithm
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