Technical Report MSC-2018-11

Title: On Polynomial time Constructions of Minimum Height Decision Tree
Authors: Waseem Makhoul
Supervisors: Nader Bshouty
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Abstract: We address the problem of constructing a minimum height decision tree of a class C in polynomial time. This problem has many interesting applications that include, to name a few, computer vision, group testing, exact learning from membership queries, and game theory.

We further study the combinatorial measure, the extended teaching dimension, ETD(C) of a class C. We show an algorithm that achieves a ETD(C)-approximation of the optimal height. When the extended dimension is small, this approximation is better than the log(|C|)-approximation known from the literature. We also show that there is no better approximation ratio unless P=NP.

We then apply our results to learning the class of disjunctions of predicates from membership queries, Bshouty et al. (2017). We show that the extended teaching dimension of this class is bounded from above by the degree d of its Hasse diagram. This gives optimal algorithms when the degree is constant.

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