44,415 research outputs found
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
A Simple FPTAS for Counting Edge Covers
An edge cover of a graph is a set of edges such that every vertex has at
least an adjacent edge in it. Previously, approximation algorithm for counting
edge covers is only known for 3 regular graphs and it is randomized. We design
a very simple deterministic fully polynomial-time approximation scheme (FPTAS)
for counting the number of edge covers for any graph. Our main technique is
correlation decay, which is a powerful tool to design FPTAS for counting
problems. In order to get FPTAS for general graphs without degree bound, we
make use of a stronger notion called computationally efficient correlation
decay, which is introduced in [Li, Lu, Yin SODA 2012].Comment: To appear in SODA 201
Finding long cycles in graphs
We analyze the problem of discovering long cycles inside a graph. We propose
and test two algorithms for this task. The first one is based on recent
advances in statistical mechanics and relies on a message passing procedure.
The second follows a more standard Monte Carlo Markov Chain strategy. Special
attention is devoted to Hamiltonian cycles of (non-regular) random graphs of
minimal connectivity equal to three
The number of matchings in random graphs
We study matchings on sparse random graphs by means of the cavity method. We
first show how the method reproduces several known results about maximum and
perfect matchings in regular and Erdos-Renyi random graphs. Our main new result
is the computation of the entropy, i.e. the leading order of the logarithm of
the number of solutions, of matchings with a given size. We derive both an
algorithm to compute this entropy for an arbitrary graph with a girth that
diverges in the large size limit, and an analytic result for the entropy in
regular and Erdos-Renyi random graph ensembles.Comment: 17 pages, 6 figures, to be published in Journal of Statistical
Mechanic
Computing a k-sparse n-length Discrete Fourier Transform using at most 4k samples and O(k log k) complexity
Given an -length input signal \mbf{x}, it is well known that its
Discrete Fourier Transform (DFT), \mbf{X}, can be computed in
complexity using a Fast Fourier Transform (FFT). If the spectrum \mbf{X} is
exactly -sparse (where ), can we do better? We show that
asymptotically in and , when is sub-linear in (precisely, where ), and the support of the non-zero DFT
coefficients is uniformly random, we can exploit this sparsity in two
fundamental ways (i) {\bf {sample complexity}}: we need only
deterministically chosen samples of the input signal \mbf{x} (where
when ); and (ii) {\bf {computational complexity}}: we can
reliably compute the DFT \mbf{X} using operations, where the
constants in the big Oh are small and are related to the constants involved in
computing a small number of DFTs of length approximately equal to the sparsity
parameter . Our algorithm succeeds with high probability, with the
probability of failure vanishing to zero asymptotically in the number of
samples acquired, .Comment: 36 pages, 15 figures. To be presented at ISIT-2013, Istanbul Turke
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