Chapter

Automata, Languages and Programming

Volume 6198 of the series Lecture Notes in Computer Science pp 690-701

Approximation Algorithms for Optimal Decision Trees and Adaptive TSP Problems

  • Anupam GuptaAffiliated withComputer Science Department, Carnegie Mellon University
  • , Viswanath NagarajanAffiliated withIBM T.J. Watson Research Center
  • , R. RaviAffiliated withTepper School of Business, Carnegie Mellon University

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(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.