1,499 research outputs found
Optimal Parallel Construction of Hamiltonian Cycles and Spanning Trees in Random Graphs
We give tight bounds on the parallel complexity of some problems involving random graphs. Specifically, we show that a Hamiltonian cycle, a breadth first spanning tree, and a maximal matching can all be constructed in \Theta(log n) expected time using n= log n processors on the CRCW PRAM. This is a substantial improvement over the best previous algorithms, which required \Theta((log log n) 2 ) time and n log 2 n processors. We then introduce a technique which allows us to prove that constructing an edge cover of a random graph from its adjacency matrix requires \Omega\Gammaequ n) expected time on a CRCW PRAM with O(n) processors. Constructing an edge cover is implicit in constructing a spanning tree, a Hamiltonian cycle, and a maximal matching, so this lower bound holds for all these problems, showing that our algorithms are optimal. This new lower bound technique is one of the very few lower bound techniques known which apply to randomized CRCW PRAM algorithms, and it pro..
Approximation Algorithms for the Asymmetric Traveling Salesman Problem : Describing two recent methods
The paper provides a description of the two recent approximation algorithms
for the Asymmetric Traveling Salesman Problem, giving the intuitive description
of the works of Feige-Singh[1] and Asadpour et.al\ [2].\newline [1] improves
the previous approximation algorithm, by improving the constant
from 0.84 to 0.66 and modifying the work of Kaplan et. al\ [3] and also shows
an efficient reduction from ATSPP to ATSP. Combining both the results, they
finally establish an approximation ratio of for ATSPP,\ considering a small ,\ improving the
work of Chekuri and Pal.[4]\newline Asadpour et.al, in their seminal work\ [2],
gives an randomized algorithm for
the ATSP, by symmetrizing and modifying the solution of the Held-Karp
relaxation problem and then proving an exponential family distribution for
probabilistically constructing a maximum entropy spanning tree from a spanning
tree polytope and then finally defining the thin-ness property and transforming
a thin spanning tree into an Eulerian walk.\ The optimization methods used in\
[2] are quite elegant and the approximation ratio could further be improved, by
manipulating the thin-ness of the cuts.Comment: 12 page
Bounds on the maximum multiplicity of some common geometric graphs
We obtain new lower and upper bounds for the maximum multiplicity of some
weighted and, respectively, non-weighted common geometric graphs drawn on n
points in the plane in general position (with no three points collinear):
perfect matchings, spanning trees, spanning cycles (tours), and triangulations.
(i) We present a new lower bound construction for the maximum number of
triangulations a set of n points in general position can have. In particular,
we show that a generalized double chain formed by two almost convex chains
admits {\Omega}(8.65^n) different triangulations. This improves the bound
{\Omega}(8.48^n) achieved by the double zig-zag chain configuration studied by
Aichholzer et al.
(ii) We present a new lower bound of {\Omega}(12.00^n) for the number of
non-crossing spanning trees of the double chain composed of two convex chains.
The previous bound, {\Omega}(10.42^n), stood unchanged for more than 10 years.
(iii) Using a recent upper bound of 30^n for the number of triangulations,
due to Sharir and Sheffer, we show that n points in the plane in general
position admit at most O(68.62^n) non-crossing spanning cycles.
(iv) We derive lower bounds for the number of maximum and minimum weighted
geometric graphs (matchings, spanning trees, and tours). We show that the
number of shortest non-crossing tours can be exponential in n. Likewise, we
show that both the number of longest non-crossing tours and the number of
longest non-crossing perfect matchings can be exponential in n. Moreover, we
show that there are sets of n points in convex position with an exponential
number of longest non-crossing spanning trees. For points in convex position we
obtain tight bounds for the number of longest and shortest tours. We give a
combinatorial characterization of the longest tours, which leads to an O(nlog
n) time algorithm for computing them
Computational Complexity for Physicists
These lecture notes are an informal introduction to the theory of
computational complexity and its links to quantum computing and statistical
mechanics.Comment: references updated, reprint available from
http://itp.nat.uni-magdeburg.de/~mertens/papers/complexity.shtm
An extensive English language bibliography on graph theory and its applications, supplement 1
Graph theory and its applications - bibliography, supplement
Pseudo-random graphs
Random graphs have proven to be one of the most important and fruitful
concepts in modern Combinatorics and Theoretical Computer Science. Besides
being a fascinating study subject for their own sake, they serve as essential
instruments in proving an enormous number of combinatorial statements, making
their role quite hard to overestimate. Their tremendous success serves as a
natural motivation for the following very general and deep informal questions:
what are the essential properties of random graphs? How can one tell when a
given graph behaves like a random graph? How to create deterministically graphs
that look random-like? This leads us to a concept of pseudo-random graphs and
the aim of this survey is to provide a systematic treatment of this concept.Comment: 50 page
- …