Approximation Algorithms for Optimal Decision Trees and Adaptive TSP Problems

TL;DR

This paper presents tight approximation algorithms for optimal decision trees and the adaptive TSP problem, significantly advancing the understanding of their computational complexity and providing near-optimal solutions.

Contribution

It introduces the first tight $O( ext{log } m)$-approximation for optimal decision trees and the first poly-logarithmic approximation for the adaptive TSP, establishing their computational bounds.

Findings

Optimal decision tree problem has a tight $O( ext{log } m)$-approximation.

Adaptive TSP admits the first poly-logarithmic approximation.

Results are tight unless improvements are made for the group Steiner tree problem.

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.