# A Tight Analysis of Greedy Yields Subexponential Time Approximation for Uniform Decision Tree

## TL;DR

This paper provides a tight analysis of the greedy algorithm for Uniform Decision Tree, showing its approximation ratio depends on the optimal cost, and introduces subexponential algorithms with implications for complexity theory.

## Contribution

It establishes a precise approximation bound for greedy algorithms on Uniform Decision Tree and introduces subexponential algorithms, resolving a conjecture and connecting to Min Sum Set Cover.

## Key findings

- Greedy algorithm achieves an $O(rac{ ext{log } n}{ ext{log } C_{OPT}})$ approximation.
- Subexponential time algorithms with ratio $rac{9.01}{ ext{alpha}}$ for all $ ext{alpha} ext{ in}(0,1)$.
- Achieving super-constant approximation ratios is unlikely to be NP-hard under ETH.

## Abstract

Decision Tree is a classic formulation of active learning: given $n$ hypotheses with nonnegative weights summing to 1 and a set of tests that each partition the hypotheses, output a decision tree using the provided tests that uniquely identifies each hypothesis and has minimum (weighted) average depth. Previous works showed that the greedy algorithm achieves a $O(\log n)$ approximation ratio for this problem and it is NP-hard beat a $O(\log n)$ approximation, settling the complexity of the problem. However, for Uniform Decision Tree, i.e. Decision Tree with uniform weights, the story is more subtle. The greedy algorithm's $O(\log n)$ approximation ratio was the best known, but the largest approximation ratio known to be NP-hard is $4-\varepsilon$. We prove that the greedy algorithm gives a $O(\frac{\log n}{\log C_{OPT}})$ approximation for Uniform Decision Tree, where $C_{OPT}$ is the cost of the optimal tree and show this is best possible for the greedy algorithm. As a corollary, we resolve a conjecture of Kosaraju, Przytycka, and Borgstrom. Leveraging this result, for all $\alpha\in(0,1)$, we exhibit a $\frac{9.01}{\alpha}$ approximation algorithm to Uniform Decision Tree running in subexponential time $2^{\tilde O(n^\alpha)}$. As a corollary, achieving any super-constant approximation ratio on Uniform Decision Tree is not NP-hard, assuming the Exponential Time Hypothesis. This work therefore adds approximating Uniform Decision Tree to a small list of natural problems that have subexponential time algorithms but no known polynomial time algorithms. All our results hold for Decision Tree with weights not too far from uniform. A key technical contribution of our work is showing a connection between greedy algorithms for Uniform Decision Tree and for Min Sum Set Cover.