Constant-Factor Approximation for the Uniform Decision Tree

• URL: http://arxiv.org/abs/2604.12036v1
• Date: Mon, 13 Apr 2026 20:24:58 GMT
• Title: Constant-Factor Approximation for the Uniform Decision Tree
• Abstract summary: We use a decomposition technique known from problems related to textsc Clustering [SODA '17, WALCOM '26], which allows us to decompose the optimal decision tree into a series of objects called subfamilies.<n>We obtain a good approximation for the textscSeparating Subfamily problem, which enables the design of the approximation algorithm for the original problem.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We resolve a long-standing open question, about the existence of a constant-factor approximation algorithm for the average-case \textsc{Decision Tree} problem with uniform probability distribution over the hypotheses. We answer the question in the affirmative by providing a simple polynomial-time algorithm with approximation ratio of $\frac{2}{1-\sqrt{(e+1)/(2e)}}+ε<11.57$. This improves upon the currently best-known, greedy algorithm which achieves $O(\log n/{\log\log n})$-approximation. The first key ingredient in our analysis is the usage of a decomposition technique known from problems related to \textsc{Hierarchical Clustering} [SODA '17, WALCOM '26], which allows us to decompose the optimal decision tree into a series of objects called separating subfamilies. The second crucial idea is to reduce the subproblem of finding a \textsc{Separating Subfamily} to an instance of the \textsc{Maximum Coverage} problem. To do so, we analyze the properties of cutting cliques into small pieces, which represent pairs of hypotheses to be separated. This allows us to obtain a good approximation for the \textsc{Separating Subfamily} problem, which then enables the design of the approximation algorithm for the original problem.

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