152 research outputs found
The complexity of two graph orientation problems
This is the post-print version of the Article. The official published version can be accessed from the link below - Copyright @ 2012 ElsevierWe consider two orientation problems in a graph, namely the minimization of the sum of all the shortest path lengths and the minimization of the diameter. Our main result is that for each positive integer k, there is a linear-time algorithm that decides for a planar graph Gwhether there is an orientation for which the diameter is at most k. We also extend this result from planar graphs to any minor-closed family F not containing all apex graphs. In contrast, it is known to be NP-complete to decide whether a graph has an orientation such that the sum of all the shortest path lengths is at most an integer specified in the input. We give a simpler proof of this result.This work is partially supported by EC Marie Curie programme NET-ACE (MEST-CT-2004-6724), and Heilbronn Institute for Mathematical Research, Bristol
Counting Euler Tours in Undirected Bounded Treewidth Graphs
We show that counting Euler tours in undirected bounded tree-width graphs is
tractable even in parallel - by proving a upper bound. This is in
stark contrast to #P-completeness of the same problem in general graphs.
Our main technical contribution is to show how (an instance of) dynamic
programming on bounded \emph{clique-width} graphs can be performed efficiently
in parallel. Thus we show that the sequential result of Espelage, Gurski and
Wanke for efficiently computing Hamiltonian paths in bounded clique-width
graphs can be adapted in the parallel setting to count the number of
Hamiltonian paths which in turn is a tool for counting the number of Euler
tours in bounded tree-width graphs. Our technique also yields parallel
algorithms for counting longest paths and bipartite perfect matchings in
bounded-clique width graphs.
While establishing that counting Euler tours in bounded tree-width graphs can
be computed by non-uniform monotone arithmetic circuits of polynomial degree
(which characterize ) is relatively easy, establishing a uniform
bound needs a careful use of polynomial interpolation.Comment: 17 pages; There was an error in the proof of the GapL upper bound
claimed in the previous version which has been subsequently remove
Revisiting Radius, Diameter, and all Eccentricity Computation in Graphs through Certificates
We introduce notions of certificates allowing to bound eccentricities in a
graph. In particular , we revisit radius (minimum eccentricity) and diameter
(maximum eccentricity) computation and explain the efficiency of practical
radius and diameter algorithms by the existence of small certificates for
radius and diameter plus few additional properties. We show how such
computation is related to covering a graph with certain balls or complementary
of balls. We introduce several new algorithmic techniques related to
eccentricity computation and propose algorithms for radius, diameter and all
eccentricities with theoretical guarantees with respect to certain graph
parameters. This is complemented by experimental results on various real-world
graphs showing that these parameters appear to be low in practice. We also
obtain refined results in the case where the input graph has low doubling
dimension, has low hyperbolicity, or is chordal
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