Decision Trees for Function Evaluation - Simultaneous Optimization of Worst and Expected Cost

TL;DR

This paper introduces a decision tree algorithm for function evaluation that optimizes both worst-case and expected costs simultaneously, achieving near-optimal performance under standard complexity assumptions.

Contribution

It presents a novel algorithm that constructs decision trees with logarithmic approximation guarantees for both worst and expected costs, a significant improvement over prior methods.

Findings

Achieves logarithmic approximation for worst-case cost

Achieves logarithmic approximation for expected cost

Optimal under the assumption that P ≠ NP

Abstract

In several applications of automatic diagnosis and active learning a central problem is the evaluation of a discrete function by adaptively querying the values of its variables until the values read uniquely determine the value of the function. In general, the process of reading the value of a variable might involve some cost, computational or even a fee to be paid for the experiment required for obtaining the value. This cost should be taken into account when deciding the next variable to read. The goal is to design a strategy for evaluating the function incurring little cost (in the worst case or in expectation according to a prior distribution on the possible variables' assignments). Our algorithm builds a strategy (decision tree) which attains a logarithmic approxima- tion simultaneously for the expected and worst cost spent. This is best possible under the assumption that $P \neq NP.$