Diagnosis determination: decision trees optimizing simultaneously worst and expected testing cost

In several applications of automatic diagnosis and active learning a central problem is the eval- uation of a discrete function by adaptively query- ing the values of its variables until the values read uniquely determine the value of the function. In general reading the value of a variable is done at the expense of some cost (computational or pos- sibly a fee to pay the corresponding experiment). 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 approximation simul- taneously for the expected and worst cost spent. This is best possible since, under standard com- plexity assumption, no algorithm can guarantee o(log n) approximation.