3,827 research outputs found
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
Improving Christofides' Algorithm for the s-t Path TSP
We present a deterministic (1+sqrt(5))/2-approximation algorithm for the s-t
path TSP for an arbitrary metric. Given a symmetric metric cost on n vertices
including two prespecified endpoints, the problem is to find a shortest
Hamiltonian path between the two endpoints; Hoogeveen showed that the natural
variant of Christofides' algorithm is a 5/3-approximation algorithm for this
problem, and this asymptotically tight bound in fact has been the best
approximation ratio known until now. We modify this algorithm so that it
chooses the initial spanning tree based on an optimal solution to the Held-Karp
relaxation rather than a minimum spanning tree; we prove this simple but
crucial modification leads to an improved approximation ratio, surpassing the
20-year-old barrier set by the natural Christofides' algorithm variant. Our
algorithm also proves an upper bound of (1+sqrt(5))/2 on the integrality gap of
the path-variant Held-Karp relaxation. The techniques devised in this paper can
be applied to other optimization problems as well: these applications include
improved approximation algorithms and improved LP integrality gap upper bounds
for the prize-collecting s-t path problem and the unit-weight graphical metric
s-t path TSP.Comment: 31 pages, 5 figure
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
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