12 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
Constant Amortized Time Enumeration of Eulerian trails
In this paper, we consider enumeration problems for edge-distinct and
vertex-distinct Eulerian trails. Here, two Eulerian trails are
\emph{edge-distinct} if the edge sequences are not identical, and they are
\emph{vertex-distinct} if the vertex sequences are not identical. As the main
result, we propose optimal enumeration algorithms for both problems, that is,
these algorithm runs in total time, where is the number of
solutions. Our algorithms are based on the reverse search technique introduced
by [Avis and Fukuda, DAM 1996], and the push out amortization technique
introduced by [Uno, WADS 2015]
Algorithmic Problems Arising in Posets and Permutations
Partially ordered sets and permutations are combinatorial structures having vast applications in theoretical computer science. In this thesis, we study various computational and algorithmic problems related to these structures. The first chapter of the thesis contains discussion about randomized fully polynomial approximation schemes obtained by employing Markov chain Monte Carlo. In this chapter we study various Markov chains that we call: the gladiator chain, the interval chain, and cube shuffling. Our objective is to identify some conditions that assure rapid mixing; and we obtain partial results. The gladiator chain is a biased random walk on the set of permutations. This chain is related to self organizing lists, and various versions of it have been studied. The interval chain is a random walk on the set of points in whose coordinates respect a partial order. Since the sample space of the interval chain is continuous, many mixing techniques for discrete chains are not applicable to it. The cube shuffle chain is a generalization of H\r{a}stad\u27s square shuffle. The importance of this chain is that it mixes in constant number of steps. In the second chapter, we are interested in calculating expected value of real valued function on a set of combinatorial structures , given a probability distribution on it. We first suggest a Markov chain Monte Carlo approach to this problem. We identify the conditions under which our proposed solution will be efficient, and present examples where it fails. Then, we study homomesy. Homomesy is a phenomenon introduced by Jim Propp and Tom Roby. We say the triple ( is a permutation mapping to itself) exhibits homomesy, if the average of along all -orbits of is a constant only depending on and . We study homomesy and obtain some results when is the set of ideals in a class of simply described lattices
Counting and sampling problems on Eulerian graphs
In this thesis we consider two sets of combinatorial structures defined on an Eulerian
graph: the Eulerian orientations and Euler tours. We are interested in the computational
problems of counting (computing the number of elements in the set) and sampling
(generating a random element of the set). Specifically, we are interested in the question
of when there exists an efficient algorithm for counting or sampling the elements of
either set.
The Eulerian orientations of a number of classes of planar lattices are of practical
significance as they correspond to configurations of certain models studied in statistical
physics. In 1992 Mihail and Winkler showed that counting Eulerian orientations of a
general Eulerian graph is #P-complete and demonstrated that the problem of sampling
an Eulerian orientation can be reduced to the tractable problem of sampling a perfect
matching of a bipartite graph. We present a proof that this problem remains #Pcomplete
when the input is restricted to being a planar graph, and analyse a natural
algorithm for generating random Eulerian orientations of one of the afore-mentioned
planar lattices. Moreover, we make some progress towards classifying the range of
planar graphs on which this algorithm is rapidly mixing by exhibiting an infinite class
of planar graphs for which the algorithm will always take an exponential amount of
time to converge.
The problem of counting the Euler tours of undirected graphs has proven to be less
amenable to analysis than that of Eulerian orientations. Although it has been known
for many years that the number of Euler tours of any directed graph can be computed in
polynomial time, until recently very little was known about the complexity of counting
Euler tours of an undirected graph. Brightwell and Winkler showed that this problem is
#P-complete in 2005 and, apart from a few very simple examples, e.g., series-parellel
graphs, there are no known tractable cases, nor are there any good reasons to believe
the problem to be intractable. Moreover, despite several unsuccessful attempts, there
has been no progress made on the question of approximability. Indeed, this problem
was considered to be one of the more difficult open problems in approximate counting
since long before the complexity of exact counting was resolved. By considering a
randomised input model, we are able to show that a very simple algorithm can sample
or approximately count the Euler tours of almost every d-in/d-out directed graph in
expected polynomial time. Then, we present some partial results towards showing that
this algorithm can be used to sample or approximately count the Euler tours of almost
every 2d-regular graph in expected polynomial time. We also provide some empirical
evidence to support the unproven conjecture required to obtain this result. As a sideresult
of this work, we obtain an asymptotic characterisation of the distribution of the
number of Eulerian orientations of a random 2d-regular graph