2,365 research outputs found
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
Approximating the Regular Graphic TSP in near linear time
We present a randomized approximation algorithm for computing traveling
salesperson tours in undirected regular graphs. Given an -vertex,
-regular graph, the algorithm computes a tour of length at most
, with high probability, in time. This improves upon a recent result by Vishnoi (\cite{Vishnoi12}, FOCS
2012) for the same problem, in terms of both approximation factor, and running
time. The key ingredient of our algorithm is a technique that uses
edge-coloring algorithms to sample a cycle cover with cycles with
high probability, in near linear time.
Additionally, we also give a deterministic
factor approximation algorithm
running in time .Comment: 12 page
Stochastic Vehicle Routing with Recourse
We study the classic Vehicle Routing Problem in the setting of stochastic
optimization with recourse. StochVRP is a two-stage optimization problem, where
demand is satisfied using two routes: fixed and recourse. The fixed route is
computed using only a demand distribution. Then after observing the demand
instantiations, a recourse route is computed -- but costs here become more
expensive by a factor lambda.
We present an O(log^2 n log(n lambda))-approximation algorithm for this
stochastic routing problem, under arbitrary distributions. The main idea in
this result is relating StochVRP to a special case of submodular orienteering,
called knapsack rank-function orienteering. We also give a better approximation
ratio for knapsack rank-function orienteering than what follows from prior
work. Finally, we provide a Unique Games Conjecture based omega(1) hardness of
approximation for StochVRP, even on star-like metrics on which our algorithm
achieves a logarithmic approximation.Comment: 20 Pages, 1 figure Revision corrects the statement and proof of
Theorem 1.
Faster Worst Case Deterministic Dynamic Connectivity
We present a deterministic dynamic connectivity data structure for undirected
graphs with worst case update time and constant query time. This improves on the previous best
deterministic worst case algorithm of Frederickson (STOC 1983) and Eppstein
Galil, Italiano, and Nissenzweig (J. ACM 1997), which had update time
. All other algorithms for dynamic connectivity are either
randomized (Monte Carlo) or have only amortized performance guarantees
Connectivity Oracles for Graphs Subject to Vertex Failures
We introduce new data structures for answering connectivity queries in graphs
subject to batched vertex failures. A deterministic structure processes a batch
of failed vertices in time and thereafter
answers connectivity queries in time. It occupies space . We develop a randomized Monte Carlo version of our data structure
with update time , query time , and space
for any failure bound . This is the first connectivity oracle for
general graphs that can efficiently deal with an unbounded number of vertex
failures.
We also develop a more efficient Monte Carlo edge-failure connectivity
oracle. Using space , edge failures are processed in time and thereafter, connectivity queries are answered in
time, which are correct w.h.p.
Our data structures are based on a new decomposition theorem for an
undirected graph , which is of independent interest. It states that
for any terminal set we can remove a set of
vertices such that the remaining graph contains a Steiner forest for with
maximum degree
On the class of graphs with strong mixing properties
We study three mixing properties of a graph: large algebraic connectivity,
large Cheeger constant (isoperimetric number) and large spectral gap from 1 for
the second largest eigenvalue of the transition probability matrix of the
random walk on the graph. We prove equivalence of this properties (in some
sense). We give estimates for the probability for a random graph to satisfy
these properties. In addition, we present asymptotic formulas for the numbers
of Eulerian orientations and Eulerian circuits in an undirected simple graph
COSMOGRAIL: the COSmological MOnitoring of GRAvItational Lenses XIII: Time delays and 9-yr optical monitoring of the lensed quasar RX J1131-1231
We present the results from nine years of optically monitoring the
gravitationally lensed z=0.658 quasar RX J1131-1231. The R-band light curves of
the four individual images of the quasar were obtained using deconvolution
photometry for a total of 707 epochs. Several sharp quasar variability features
strongly constrain the time delays between the quasar images. Using three
different numerical techniques, we measure these delays for all possible pairs
of quasar images while always processing the four light curves simultaneously.
For all three methods, the delays between the three close images A, B, and C
are compatible with being 0, while we measure the delay of image D to be 91
days, with a fractional uncertainty of 1.5% (1 sigma), including systematic
errors. Our analysis of random and systematic errors accounts in a realistic
way for the observed quasar variability, fluctuating microlensing magnification
over a broad range of temporal scales, noise properties, and seasonal gaps.
Finally, we find that our time-delay measurement methods yield compatible
results when applied to subsets of the data.Comment: 11 pages, 9 figures, minor additions to the text only, techniques and
results remain unchanged, A&A in pres
- …