Selecting the Kth largest-area convex polygon

Abstract

The purpose of this note is to solve an open problem submitted by B. Chazelle [2]: Given a set P of n points in the Euclidean plane, select the kth largest-area convex polygon determined by subsets of P. We show that the decision problem is NP-hard by a reduction from the problem of finding the kth largest m-tuple [8] determined by m sets X1,X2,…,Xm. We also show that the enumeration problem is #P-complete and exhibit a pseudo-polynomial-time algorithm for the decision problem.