5,167 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
Network Design with Coverage Costs
We study network design with a cost structure motivated by redundancy in data
traffic. We are given a graph, g groups of terminals, and a universe of data
packets. Each group of terminals desires a subset of the packets from its
respective source. The cost of routing traffic on any edge in the network is
proportional to the total size of the distinct packets that the edge carries.
Our goal is to find a minimum cost routing. We focus on two settings. In the
first, the collection of packet sets desired by source-sink pairs is laminar.
For this setting, we present a primal-dual based 2-approximation, improving
upon a logarithmic approximation due to Barman and Chawla (2012). In the second
setting, packet sets can have non-trivial intersection. We focus on the case
where each packet is desired by either a single terminal group or by all of the
groups, and the graph is unweighted. For this setting we present an O(log
g)-approximation.
Our approximation for the second setting is based on a novel spanner-type
construction in unweighted graphs that, given a collection of g vertex subsets,
finds a subgraph of cost only a constant factor more than the minimum spanning
tree of the graph, such that every subset in the collection has a Steiner tree
in the subgraph of cost at most O(log g) that of its minimum Steiner tree in
the original graph. We call such a subgraph a group spanner.Comment: Updated version with additional result
Network Design Problems with Bounded Distances via Shallow-Light Steiner Trees
In a directed graph with non-correlated edge lengths and costs, the
\emph{network design problem with bounded distances} asks for a cost-minimal
spanning subgraph subject to a length bound for all node pairs. We give a
bi-criteria -approximation for this
problem. This improves on the currently best known linear approximation bound,
at the cost of violating the distance bound by a factor of at
most~.
In the course of proving this result, the related problem of \emph{directed
shallow-light Steiner trees} arises as a subproblem. In the context of directed
graphs, approximations to this problem have been elusive. We present the first
non-trivial result by proposing a
-ap\-proxi\-ma\-tion, where are the
terminals.
Finally, we show how to apply our results to obtain an
-approximation for
\emph{light-weight directed -spanners}. For this, no non-trivial
approximation algorithm has been known before. All running times depends on
and and are polynomial in for any fixed
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