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A decision tree recursively splits a feature space $\mathbb{R}^{d}$ and thenassigns class labels based on the resulting partition. Decision trees have beenpart of the basic machine-learning toolkit for decades. A large body of worktreats heuristic algorithms to compute a decision tree from training data,usually aiming to minimize in particular the size of the resulting tree. Incontrast, little is known about the complexity of the underlying computationalproblem of computing a minimum-size tree for the given training data. We studythis problem with respect to the number $d$ of dimensions of the feature space.We show that it can be solved in $O(n^{2d + 1}d)$ time, but under reasonablecomplexity-theoretic assumptions it is not possible to achieve $f(d) \cdotn^{o(d / \log d)}$ running time, where $n$ is the number of training examples.The problem is solvable in $(dR)^{O(dR)} \cdot n^{1+o(1)}$ time, if there areexactly two classes and $R$ is an upper bound on the number of tree leaveslabeled with the first~class.

The Influence of Dimensions on the Complexity of Computing Decision Trees