Simpler Logarithmic Approximation Algorithms for the Optimal Decision Tree and Adaptive Set Cover

Abstract

We study a well-known task of constructing a decision tree identifying an unknown hypothesis from a given ground set of hypotheses under both the average- and worst-case cost. The Optimal Decision Tree problem has been extensively studied in the literature and $\mathcal{O}(\log n)$-approximation guarantees are known for both cost criteria (here $n$ is the number of hypotheses). Although the algorithms achieving this approximation ratio are usually relatively simple, their analysis often turns out to be quite technical. Hereby, we show a new algorithm with a simplified analysis which simultaneously achieves an $\mathcal{O}(\log n)$-approximation for both cost criteria. Moreover, the leading constant under the $\mathcal{O}$-notation in the approximation ratio is relatively small, below $3.65$ for the worst-case cost and twice as much for the average-case cost, assuming base $2$ logarithm. Our algorithm for the Optimal Decision Tree behaves greedily with respect to the hereby introduced Separating Subfamily problem which asks for the cheapest subfamily of tests which partitions the hypotheses into pieces of small enough size. We show that the Separating Subfamily can itself be reduced to an instance of the well-known Maximum Coverage problem. At the core of our approach lies exploiting properties of cutting a clique into small pieces, where edges represent pairs of hypotheses to be separated. As an application of our result we also provide an $\mathcal{O}(\log (k\cdot n))$-approximation for the Adaptive Set Cover problem, where $k$ is the number of possible realizations.