Reasons to study combinatorial methods

There are generally two good reasons to study combinatorial approaches to motion planning:

1. In many applications, one may only be interested in a special class of planning problems. For example, the world might be 2D, and the robot might only be capable of translation. For many special classes, elegant and efficient algorithms can be developed. These algorithms are complete, do not depend on approximation, and can offer much better performance than sampling-based planning methods, such as those in Chapter 5.
2. It is both interesting and satisfying to know that there are complete algorithms for an extremely broad class of motion planning problems. Thus, even if the class of interest does not have some special limiting assumptions, there still exist general-purpose tools and algorithms that can solve it. These algorithms also provide theoretical upper bounds on the time needed to solve motion planning problems.