Approximation algorithms for optimal decision trees and adaptive TSP problems

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(logm)-approximation algorithm. We also consider a more substantial generalization, the Adaptive TSP problem, which can be used to model switching costs between tests in the optimal decision tree 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 adaptive TSP 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.