1 research outputs found
A Tight Analysis of Greedy Yields Subexponential Time Approximation for Uniform Decision Tree
Decision Tree is a classic formulation of active learning: given
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
approximation ratio for this problem and it is NP-hard beat a
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 approximation
ratio was the best known, but the largest approximation ratio known to be
NP-hard is . We prove that the greedy algorithm gives a
approximation for Uniform Decision Tree, where
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 , we
exhibit a approximation algorithm to Uniform Decision
Tree running in subexponential time . 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.Comment: 40 pages, 5 figure