Counting Triangulations and other Crossing-free Structures via Onion Layers

Let P be a set of n points in the plane. A crossing-free structure on P is a plane graph with vertex set P . Examples of crossing-free structures include triangulations of P , spanning cycles of P , also known as polygonalizations of P , among others. There has been a large amount of research trying to bound the number of such structures. In particular, bounding the number of (straight-edge) triangulations spanned by P has received considerable attention. It is currently known that every set of n points has at most O(30) and at least Ω(2.43) triangulations. However, much less is known about the algorithmic problem of counting crossing-free structures of a given set P . For example, no algorithm for counting triangulations is known that, on all instances, performs faster than enumerating all triangulations. In this paper we develop a general technique for computing the number of crossing-free structures of an input set P . We apply the technique to obtain algorithms for computing the number of triangulations, matchings, and spanning cycles of P . The running time of our algorithms is upper bounded by n, where k is the number of onion layers of P . In particular, for k = O(1) our algorithms run in polynomial time. In addition, we show that our algorithm for counting triangulations is never slower than O(3.1414), even when k = Θ(n). Given that there are several well-studied configurations of points with at least Ω(3.464) triangulations, and some even with Ω(8) triangulations, our algorithm asymptotically outperforms any enumeration algorithm for such instances, and it has better worst-case behavior than the recent algorithm shown in [1], which also beats enumeration in those instances. In fact, it is widely believed that any set of n points must have at least Ω(3.464) triangulations. If this is true, then our algorithm is strictly sub-linear in the number of triangulations counted. We also show that our techniques are general enough to solve the Restricted-Triangulation-Counting-Problem, which we prove to be W [2]-hard in the parameter k. This implies a “no free lunch” result: In order to be fixed-parameter tractable, our general algorithm must rely on additional properties that are specific to the considered class of structures. Fachrichtung Informatik, Universität des Saarlandes, [email protected]. Partially Supported by CONACYT-DAAD of México. Max-Planck-Institut für Informatik, [email protected]. Fachrichtung Informatik, Universität des Saarlandes, [email protected] Max-Planck-Institut für Informatik. [email protected].