Approximation Algorithms for Optimal Decision Trees and Adaptive TSP Problems

We consider the problem of constructing optimal decision trees: given a collection of tests that can disambiguate between a set of m possible diseases, each test having a cost, and the a priori likelihood of any particular disease, what is a good adaptive strategy to perform these tests to minimize the expected cost to identify the disease? This problem has been studied in several works, with O(log m)-approximations known in the special cases when either costs or probabilities are uniform. In this paper, we settle the approximability of the general problem by giving a tight O(log m)-approximation algorithm. We also consider a substantial generalization, the adaptive traveling salesman problem. Given an underlying metric space, a random subset S of vertices is drawn from a known distribution, but S is initially unknown—we get information about whether any vertex is in S only when it is visited. What is a good adaptive strategy to visit all vertices in the random subset S while minimizing the expected dista...

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