Interval relations play a significant role in many resource allocation, temporal reasoning, biological and scheduling problems. We saw this in Sections 1.1 and 4.1 in our motivating examples for interval graphs, tolerance graphs and interval probe graphs. Intervals can represent events in time, which may conflict or may be compatible. They can represent certain tasks to be performed according to a timetable which must be assigned distinct processors or people. Or they may represent fragments of DNA, which are compatible or incompatible.

For many optimization problems, such as graph coloring or finding maximum stable sets, there are efficient algorithms that give solutions when the set of graphs under consideration is restricted to a structured family. Many applications reduce to solving optimization problems on such families of graphs. Indeed, at the very beginning of this book, a 4-coloring of the tolerance graph in Figure 1.3 provided an assignment of four meeting rooms for that motivating example. In a similar application, with say only one room available for a given collection of meetings (intervals) with tolerances, a maximum stable set would provide the largest number of meetings from the collection that can be scheduled. In this chapter, we investigate these algorithmic aspects of tolerance graphs.

Narasimhan and Manber (1992) were the first to study the chromatic number, clique and stable set problems for representations of tolerance graphs.