1,966 research outputs found
Constant-Factor Approximation for TSP with Disks
We revisit the traveling salesman problem with neighborhoods (TSPN) and
present the first constant-ratio approximation for disks in the plane: Given a
set of disks in the plane, a TSP tour whose length is at most times
the optimal can be computed in time that is polynomial in . Our result is
the first constant-ratio approximation for a class of planar convex bodies of
arbitrary size and arbitrary intersections. In order to achieve a
-approximation, we reduce the traveling salesman problem with disks, up
to constant factors, to a minimum weight hitting set problem in a geometric
hypergraph. The connection between TSPN and hitting sets in geometric
hypergraphs, established here, is likely to have future applications.Comment: 14 pages, 3 figure
Approximating the Held-Karp Bound for Metric TSP in Nearly Linear Time
We give a nearly linear time randomized approximation scheme for the
Held-Karp bound [Held and Karp, 1970] for metric TSP. Formally, given an
undirected edge-weighted graph on edges and , the
algorithm outputs in time, with high probability, a
-approximation to the Held-Karp bound on the metric TSP instance
induced by the shortest path metric on . The algorithm can also be used to
output a corresponding solution to the Subtour Elimination LP. We substantially
improve upon the running time achieved previously
by Garg and Khandekar. The LP solution can be used to obtain a fast randomized
-approximation for metric TSP which improves
upon the running time of previous implementations of Christofides' algorithm
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
The Unreasonable Success of Local Search: Geometric Optimization
What is the effectiveness of local search algorithms for geometric problems
in the plane? We prove that local search with neighborhoods of magnitude
is an approximation scheme for the following problems in the
Euclidian plane: TSP with random inputs, Steiner tree with random inputs,
facility location (with worst case inputs), and bicriteria -median (also
with worst case inputs). The randomness assumption is necessary for TSP
On the Number of Iterations for Dantzig-Wolfe Optimization and Packing-Covering Approximation Algorithms
We give a lower bound on the iteration complexity of a natural class of
Lagrangean-relaxation algorithms for approximately solving packing/covering
linear programs. We show that, given an input with random 0/1-constraints
on variables, with high probability, any such algorithm requires
iterations to compute a
-approximate solution, where is the width of the input.
The bound is tight for a range of the parameters .
The algorithms in the class include Dantzig-Wolfe decomposition, Benders'
decomposition, Lagrangean relaxation as developed by Held and Karp [1971] for
lower-bounding TSP, and many others (e.g. by Plotkin, Shmoys, and Tardos [1988]
and Grigoriadis and Khachiyan [1996]). To prove the bound, we use a discrepancy
argument to show an analogous lower bound on the support size of
-approximate mixed strategies for random two-player zero-sum
0/1-matrix games
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