87 research outputs found
Exact arborescences, matchings and cycles
AbstractSuppose we are given a graph in which edge has an integral weight. An âexactâ problem is to determine whether a desired structure exists for which the sum of the edge weights is exactly k for some prescribed k.We consider the special case of the problem in which all costs are zero or one for arborescences and show that a âcontinuityâ property is prossessed similar to that possessed by matroids. This enables us to determine in polynomial time the complete set of values of k for which a solution exists. We also give a minmax theorem for the maximum possible value of k, in terms of a packing of certain directed cuts in the graph.We also show how enumerative techniques can be used to solve the general exact problem for arborescences (implying spanning trees), perfect matchings in planar graphs and sets of disjoint cycles in a class of planar directed graphs which includes those of degree three. For these problems, we thereby obtain polynomial algorithms provided that the weights are bounded by a constant or encoded in unary
Sampling Arborescences in Parallel
We study the problem of sampling a uniformly random directed rooted spanning tree, also known as an arborescence, from a possibly weighted directed graph. Classically, this problem has long been known to be polynomial-time solvable; the exact number of arborescences can be computed by a determinant [Tutte, 1948], and sampling can be reduced to counting [Jerrum et al., 1986; Jerrum and Sinclair, 1996]. However, the classic reduction from sampling to counting seems to be inherently sequential. This raises the question of designing efficient parallel algorithms for sampling. We show that sampling arborescences can be done in RNC.
For several well-studied combinatorial structures, counting can be reduced to the computation of a determinant, which is known to be in NC [Csanky, 1975]. These include arborescences, planar graph perfect matchings, Eulerian tours in digraphs, and determinantal point processes. However, not much is known about efficient parallel sampling of these structures. Our work is a step towards resolving this mystery
Bernoulli Factories for Flow-Based Polytopes
We construct explicit combinatorial Bernoulli factories for the class of
\emph{flow-based polytopes}; integral 0/1-polytopes defined by a set of network
flow constraints. This generalizes the results of Niazadeh et al. (who
constructed an explicit factory for the specific case of bipartite perfect
matchings) and provides novel exact sampling procedures for sampling paths,
circulations, and -flows. In the process, we uncover new connections to
algebraic combinatorics
Approximation Algorithms for Multi-Criteria Traveling Salesman Problems
In multi-criteria optimization problems, several objective functions have to
be optimized. Since the different objective functions are usually in conflict
with each other, one cannot consider only one particular solution as the
optimal solution. Instead, the aim is to compute a so-called Pareto curve of
solutions. Since Pareto curves cannot be computed efficiently in general, we
have to be content with approximations to them.
We design a deterministic polynomial-time algorithm for multi-criteria
g-metric STSP that computes (min{1 +g, 2g^2/(2g^2 -2g +1)} + eps)-approximate
Pareto curves for all 1/2<=g<=1. In particular, we obtain a
(2+eps)-approximation for multi-criteria metric STSP. We also present two
randomized approximation algorithms for multi-criteria g-metric STSP that
achieve approximation ratios of (2g^3 +2g^2)/(3g^2 -2g +1) + eps and (1 +g)/(1
+3g -4g^2) + eps, respectively.
Moreover, we present randomized approximation algorithms for multi-criteria
g-metric ATSP (ratio 1/2 + g^3/(1 -3g^2) + eps) for g < 1/sqrt(3)), STSP with
weights 1 and 2 (ratio 4/3) and ATSP with weights 1 and 2 (ratio 3/2). To do
this, we design randomized approximation schemes for multi-criteria cycle cover
and graph factor problems.Comment: To appear in Algorithmica. A preliminary version has been presented
at the 4th Workshop on Approximation and Online Algorithms (WAOA 2006
Local statistics of lattice dimers
We show how to compute the probability of any given local configuration in a
random tiling of the plane with dominos. That is, we explicitly compute the
measures of cylinder sets for the measure of maximal entropy on the space
of tilings of the plane with dominos.
We construct a measure on the set of lozenge tilings of the plane, show
that its entropy is the topological entropy, and compute explicitly the
-measures of cylinder sets.
As applications of these results, we prove that the translation action is
strongly mixing for and , and compute the rate of convergence to
mixing (the correlation between distant events). For the measure we
compute the variance of the height function.Comment: 27 pages, 6 figure
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
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