Approximating Optimal Decision Trees

We give a ln(n) + 1-approximation for the decision tree (DT) problem. We also show that DT does not have a PTAS unless P=NP. An instance of DT is a set of m binary tests T = (T1, . . . , Tm) and a set of n items X = (X1, . . . , Xn). The goal is to output a binary tree where each internal node is a test, each leaf is an item and the average number of tests used to uniquely identify each item (or equivalently, the total external path length) is minimized. In addition, we show DT does not have a PTAS unless P=NP. DT has a rich history in computer science with applications ranging from medical diagnosis to experiment design. Our work, while providing the first nontrivial upper and lower bounds on approximating DT, also demonstrates that DT and a subtly different problem which also bears the name decision tree have fundamentally different approximation complexity. In addition, we show a connection between ConDT and a third type of decision tree problem called MinDT, which allows us to show that no 2 δ(n)-approximation exists for MinDT, for δ < 1, unless NP is quasi-polynomial.