On polynomial time constructions of minimum height decision tree

A decision tree T in B_m := {0, 1}^m is a binary tree where each of its internal nodes is labeled with an integer in [m] = {1, 2, . . ., m}, each leaf is labeled with an assignment a ∈ B_m and each internal node has two outgoing edges that are labeled with 0 and 1, respectively. Let A ⊂ {0, 1}^m. We say that T is a decision tree for A if (1) For every a ∈ A there is one leaf of T that is labeled with a. (2) For every path from the root to a leaf with internal nodes labeled with i₁, i₂, . . ., i_k ∈ [m], a leaf labeled with a ∈ A and edges labeled with ξ_i1 , . . ., ξ_ik ∈ {0, 1}, a is the only element in A that satisfies a_ij = ξ_ij for all j = 1, . . ., k. Our goal is to write a polynomial time (in n := |A| and m) algorithm that for an input A ⊆ 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 [4, 15] gave a polynomial time (ln |A|)- approximation algorithm (for the depth). The result of Dinur and Steurer [7] for set cover implies that this problem cannot be approximated with ratio (1 − o(1)) · ln |A|, unless P=NP. Moshkov studied in [15, 13, 14] 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 exponential time ETD(A)/ log(ETD(A))-approximation algorithm and a polynomial time 2(ln 2)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) ≤ 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 then apply the above results to learning the class of disjunctions of predicates from membership queries [5]. 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.