Double Threshold Digraphs

A semiorder is a model of preference relations where each element x is associated with a utility value alpha(x), and there is a threshold t such that y is preferred to x iff alpha(y) - alpha(x) > t. These are motivated by the notion that there is some uncertainty in the utility values we assign an object or that a subject may be unable to distinguish a preference between objects whose values are close. However, they fail to model the well-known phenomenon that preferences are not always transitive. Also, if we are uncertain of the utility values, it is not logical that preference is determined absolutely by a comparison of them with an exact threshold. We propose a new model in which there are two thresholds, t_1 and t_2; if the difference alpha(y) - alpha(x) is less than t_1, then y is not preferred to x; if the difference is greater than t_2 then y is preferred to x; if it is between t_1 and t_2, then y may or may not be preferred to x. We call such a relation a (t_1,t_2) double-threshold semiorder, and the corresponding directed graph G = (V,E) a (t_1,t_2) double-threshold digraph. Every directed acyclic graph is a double-threshold digraph; increasing bounds on t_2/t_1 give a nested hierarchy of subclasses of the directed acyclic graphs. In this paper we characterize the subclasses in terms of forbidden subgraphs, and give algorithms for finding an assignment of utility values that explains the relation in terms of a given (t_1,t_2) or else produces a forbidden subgraph, and finding the minimum value lambda of t_2/t_1 that is satisfiable for a given directed acyclic graph. We show that lambda gives a useful measure of the complexity of a directed acyclic graph with respect to several optimization problems that are NP-hard on arbitrary directed acyclic graphs.