Abstract
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 in B_m and each internal node has two outgoing edges that are labeled with 0 and 1, respectively. Let A subset {0,1}^m. We say that T is a decision tree for A if (1) For every a in 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 i_1,i_2,...,i_k in[m], a leaf labeled with a in A and edges labeled with xi_{i_1},...,xi_{i_k}in {0,1}, a is the only element in A that satisfies a_{i_j}=xi_{i_j} for all j=1,...,k.
Our goal is to write a polynomial time (in n:=A and m) algorithm 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 [Esther M. Arkin et al., 1998; Mikhail Ju. Moshkov, 2004] gave a polynomial time (ln A) approximation algorithm (for the depth). The result of Dinur and Steurer [Irit Dinur and David Steurer, 2014] for set cover implies that this problem cannot be approximated with ratio (1o(1))* ln A, unless P=NP. Moshkov studied in [Mikhail Ju. Moshkov, 2004; Mikhail Ju. Moshkov, 1982; Mikhail Ju. Moshkov, 1982] 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 [Nader H. Bshouty et al., 2017]. 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.
BibTeX  Entry
@InProceedings{hbshouty_et_al:LIPIcs:2018:9982,
author = {Nader H. Bshouty and Waseem Makhoul},
title = {{On Polynomial Time Constructions of Minimum Height Decision Tree}},
booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)},
pages = {34:134:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770941},
ISSN = {18688969},
year = {2018},
volume = {123},
editor = {WenLian Hsu and DerTsai Lee and ChungShou Liao},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/9982},
URN = {urn:nbn:de:0030drops99824},
doi = {10.4230/LIPIcs.ISAAC.2018.34},
annote = {Keywords: Decision Tree, Minimal Depth, Approximation algorithms}
}
Keywords: 

Decision Tree, Minimal Depth, Approximation algorithms 
Collection: 

29th International Symposium on Algorithms and Computation (ISAAC 2018) 
Issue Date: 

2018 
Date of publication: 

06.12.2018 