Optimal ordering of independent tests with precedence constraints

We consider scenarios in which a sequence of tests is to be applied to an object; the result of a test may be that a decision (such as the classification of the object) can be made without running additional tests. Thus, one seeks an ordering of the tests that is optimal in some sense, such as minimum expected resource consumption. Sequences of tests are commonly used in computer vision (Paul A. Viola and Michael J. Jones (2001) [15]) and other applications. Finding an optimal ordering is easy when the tests are completely independent. Introducing precedence constraints, we show that the optimization problem becomes NP-hard when the constraints are given by means of a general partial order. Restrictions of the constraints to non-trivial special cases that allow for low-order polynomial-time algorithms are examined.