Complexity of the Exact Solution to the Test Sequencing Problem

Consider a doctor choosing a treatment for an uncertain disorder for which there are n costly tests available. Based on the test results observed so far, the doctor can either order another test or proceed to the treatment decision. Although test sequencing is a problem that arises frequently in many decision situations, finding an exact solution is NP-hard with respect to n. In this paper, we analyze the time complexity of classic symmetric and asymmetric formulations, using influence diagrams and decision trees, to the general test sequencing problem, making no assumptions of conditional independence among the test results. We develop an alternative influence diagram formulation that scales better, and show how a decision circuit formulation improves even more on the decision tree solution through recursive coalescence. We prove that this decision circuit formulation achieves the lower bound complexity for any method for the general test sequencing problem that examines the entire policy space. As a result, the problem is tractable for much larger n than has been possible to date.