37 research outputs found
On disjoint paths in acyclic planar graphs
We give an algorithm with complexity for the integer
multiflow problem on instances with an acyclic planar digraph
and Eulerian. Here, is a polynomial function, , and is the maximum request . When is
fixed, this gives a polynomial algorithm for the arc-disjoint paths problem
under the same hypothesis
The hardness of routing two pairs on one face
We prove the NP-completeness of the integer multiflow problem in planar
graphs, with the following restrictions: there are only two demand edges, both
lying on the infinite face of the routing graph. This was one of the open
challenges concerning disjoint paths, explicitly asked by M\"uller. It also
strengthens Schw\"arzler's recent proof of one of the open problems of
Schrijver's book, about the complexity of the edge-disjoint paths problem with
terminals on the outer boundary of a planar graph. We also give a directed
acyclic reduction. This proves that the arc-disjoint paths problem is
NP-complete in directed acyclic graphs, even with only two demand arcs
Congestion in planar graphs with demands on faces
We give an algorithm to route a multicommodity flow in a planar graph
with congestion , where is the maximum number of terminals on
the boundary of a face, when each demand edge lie on a face of . We also
show that our specific method cannot achieve a substantially better congestion
Packing and covering with balls on Busemann surfaces
In this note we prove that for any compact subset of a Busemann surface
(in particular, for any simple polygon with geodesic metric)
and any positive number , the minimum number of closed balls of radius
with centers at and covering the set is at most 19
times the maximum number of disjoint closed balls of radius centered
at points of : , where and
are the covering and the packing numbers of by -balls.Comment: 27 page
The graphs with the max-Mader-flow-min-multiway-cut property
We are given a graph , an independant set of \emph{terminals}, and a function . We want to know if the maximum -packing of vertex-disjoint paths with extremities in is equal to the minimum weight of a vertex-cut separating . We call \emph{Mader-Mengerian} the graphs with this property for each independant set and each weight function . We give a characterization of these graphs in term of forbidden minors, as well as a recognition algorithm and a simple algorithm to find maximum packing of paths and minimum multicuts in those graphs
Maximum Weight Disjoint Paths in Outerplanar Graphs via Single-Tree Cut Approximators
Since 1997 there has been a steady stream of advances for the maximum
disjoint paths problem. Achieving tractable results has usually required
focusing on relaxations such as: (i) to allow some bounded edge congestion in
solutions, (ii) to only consider the unit weight (cardinality) setting, (iii)
to only require fractional routability of the selected demands (the
all-or-nothing flow setting). For the general form (no congestion, general
weights, integral routing) of edge-disjoint paths ({\sc edp}) even the case of
unit capacity trees which are stars generalizes the maximum matching problem
for which Edmonds provided an exact algorithm. For general capacitated trees,
Garg, Vazirani, Yannakakis showed the problem is APX-Hard and Chekuri, Mydlarz,
Shepherd provided a -approximation. This is essentially the only setting
where a constant approximation is known for the general form of \textsc{edp}.
We extend their result by giving a constant-factor approximation algorithm for
general-form \textsc{edp} in outerplanar graphs. A key component for the
algorithm is to find a {\em single-tree} cut approximator for
outerplanar graphs. Previously cut approximators were only known via
distributions on trees and these were based implicitly on the results of Gupta,
Newman, Rabinovich and Sinclair for distance tree embeddings combined with
results of Anderson and Feige.Comment: 19 pages, 6 figure
Modules in Robinson Spaces
A Robinson space is a dissimilarity space (i.e., a set of size
and a dissimilarity on ) for which there exists a total order on
such that implies that .
Recognizing if a dissimilarity space is Robinson has numerous applications in
seriation and classification. An mmodule of (generalizing the notion of
a module in graph theory) is a subset of which is not distinguishable
from the outside of , i.e., the distance from any point of to
all points of is the same. If is any point of , then and
the maximal by inclusion mmodules of not containing define a
partition of , called the copoint partition. In this paper, we investigate
the structure of mmodules in Robinson spaces and use it and the copoint
partition to design a simple and practical divide-and-conquer algorithm for
recognition of Robinson spaces in optimal time
The hardness of routing two pairs on one face
International audienceWe prove the NP-completeness of the integer multiflow problem in planar graphs, with the following restrictions: there are only two demand edges, both lying on the infinite face of the routing graph. This was one of the open challenges concerning disjoint paths, explicitly asked by Müller. It also strengthens Schwärzler's recent proof of one of the open problems of Schrijver's book, about the complexity of the edge-disjoint paths problem with terminals on the outer boundary of a planar graph. We also give a directed acyclic reduction. This proves that the arc-disjoint paths problem is NP-complete in directed acyclic graphs, even with only two demand arcs