2,988 research outputs found
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
A general framework for coloring problems: old results, new results, and open problems
In this survey paper we present a general framework for coloring problems that was introduced in a joint paper which the author presented at WG2003. We show how a number of different types of coloring problems, most of which have been motivated from frequency assignment, fit into this framework. We give a survey of the existing results, mainly based on and strongly biased by joint work of the author with several different groups of coauthors, include some new results, and discuss several open problems for each of the variants
Contractions, Removals and How to Certify 3-Connectivity in Linear Time
It is well-known as an existence result that every 3-connected graph G=(V,E)
on more than 4 vertices admits a sequence of contractions and a sequence of
removal operations to K_4 such that every intermediate graph is 3-connected. We
show that both sequences can be computed in optimal time, improving the
previously best known running times of O(|V|^2) to O(|V|+|E|). This settles
also the open question of finding a linear time 3-connectivity test that is
certifying and extends to a certifying 3-edge-connectivity test in the same
time. The certificates used are easy to verify in time O(|E|).Comment: preliminary versio
- …