Double Threshold Digraphs

Abstract

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 $\alpha(y) - \alpha(x)$ 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 then $y$ may or may not be preferred to $x$. We call such a relation a double-threshold semiorder, and the corresponding directed graph $G = (V,E)$ a double threshold digraph. Every directed acyclic graph is a double threshold graph; 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 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 measure of the complexity of a directed acyclic graph with respect to several optimization problems that are NP-hard on arbitrary directed acyclic graphs.