On Polynomial time Constructions of Minimum Height Decision Tree
This provides improved approximation algorithms for decision tree construction, relevant to applications like computer vision and learning, but is incremental over prior work.
The paper tackles the problem of constructing minimum depth decision trees for a given set A, showing that a polynomial time algorithm achieves a (ln 2) DEN(A)-approximation for depth, which is tight unless P=NP, and applies this to learning disjunctions with optimal algorithms for constant-degree cases.
In this paper we study a polynomial time algorithms that for an input $A\subseteq {B_m}$ outputs a decision tree for $A$ of minimum depth. This problem has many applications that include, to name a few, computer vision, group testing, exact learning from membership queries and game theory. Arkin et al. and Moshkov gave a polynomial time $(\ln |A|)$- approximation algorithm (for the depth). The result of Dinur and Steurer for set cover implies that this problem cannot be approximated with ratio $(1-o(1))\cdot \ln |A|$, unless P=NP. Moskov the combinatorial measure of extended teaching dimension of $A$, $ETD(A)$. He showed that $ETD(A)$ is a lower bound for the depth of the decision tree for $A$ and then gave an {\it exponential time} $ETD(A)/\log(ETD(A))$-approximation algorithm. In this paper we further study the $ETD(A)$ measure and a new combinatorial measure, $DEN(A)$, that we call the density of the set $A$. We show that $DEN(A)\le ETD(A)+1$. We then give two results. The first result is that the lower bound $ETD(A)$ of Moshkov for the depth of the decision tree for $A$ is greater than the bounds that are obtained by the classical technique used in the literature. The second result is a polynomial time $(\ln 2) DEN(A)$-approximation (and therefore $(\ln 2) ETD(A)$-approximation) algorithm for the depth of the decision tree of $A$. We also show that a better approximation ratio implies P=NP. We then apply the above results to learning the class of disjunctions of predicates from membership queries. We show that the $ETD$ of this class is bounded from above by the degree $d$ of its Hasse diagram. We then show that Moshkov algorithm can be run in polynomial time and is $(d/\log d)$-approximation algorithm. This gives optimal algorithms when the degree is constant. For example, learning axis parallel rays over constant dimension space.