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Approximation Algorithms for Optimal Decision Trees and Adaptive TSP Problems.

Чт, 03/04/2010 - 16:01 — SomNambul
Authors: Anupam Gupta, Viswanath Nagarajan, R. Ravi, Ravishankar Krishnaswamy
Subjects: Data Structures and Algorithms
link: http://arxiv.org/abs/1003.0722
Abstract

We consider the problem of constructing optimal decision trees: given a
collection of tests which can disambiguate between a set of $m$ possible
diseases, each test having a cost, and the a-priori likelihood of the patient
having any particular disease, what is a good adaptive strategy to perform
these tests to minimize the expected cost to identify the disease? We settle
the approximability of this problem by giving a tight $O(\log m)$-approximation
algorithm. We also consider a more substantial generalization, the Adaptive TSP
problem. Given an underlying metric space, a random subset $S$ of cities is
drawn from a known distribution, but $S$ is initially unknown to us--we get
information about whether any city is in $S$ only when we visit the city in
question. What is a good adaptive way of visiting all the cities in the random
subset $S$ while minimizing the expected distance traveled? For this problem,
we give the first poly-logarithmic approximation, and show that this algorithm
is best possible unless we can improve the approximation guarantees for the
well-known group Steiner tree problem.

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