An empirical study on randomized optimal area polygonization of planar point sets

While random polygon generation from a set of planar points has been widely investigated in the literature, very few works address the construction of a simple polygon with minimum area (MINAP) or maximum area (MAXAP) from a set of fixed planar points. Currently, no deterministic algorithms are available to compute MINAP/MAXAP, as the problems have been shown to be NP-complete. In this article, we present a greedy heuristic for computing the approximate MINAP of any given planar point set using the technique of randomized incremental construction. For a given point set of n points, the proposed algorithm takes O(n² log n) time and O(n) space. It is rather simplistic in nature, hence very easy for implementation and maintenance. We report on various experimental studies on the behavior of a randomized heuristic on different point set instances. Test data have been taken from the SPAETH cluster data base and TSPLIB library. Experimental results indicate that the proposed algorithm outperforms its counterparts for generating a tighter upper bound on the optimal minimum area polygon for large-sized point sets.