Minimum-Perimeter Enclosing k-gon

Introduction Let P = p1, . . . , pn be a simple polygon (all polygons are assumed convex throughout this paper). A fundamental problem in geometric optimization is to compute a minimum-area or a minimum-perimeter convex k-gon (denoted QA or Qp, resp.) that encloses P . While efficient algorithms for finding QA are known for more than 20 years [8, 1, 2], the problem of finding Qp has remained open; the problem is posed as open in [3, 6, 7, 9, 12, 14, 5]. 1 Chang and Yap [8] give a comprehensive classification of the incluson/enclosure problems, but do not mention the minimum-perimeter enclosing kgon problem (Enc(Pall, Pk, perimeter), in their terminology) at all. We give the first polynomial-time algorithms for computing Qp. In order to obtain our solution, we prove a structural result about an optimal polygon: Local optimality implies that it is “flush” with P (Lemma 1). As a by-product we obtain an algorithm for finding the minimum-perimeter “envelope” — a convex k-gon with a specified sequence of interior angles. Our proofs are very simple and are based on elementary geometry.2 The exact coordinates of the vertices of Qp are given by the roots of high-degree polynomials. In general, it is impossible to find the coordinates exactly in polynomial time [4]. Thus, given ε > 0 as a part of the input to the problem, we will be satisfied with a (1 + ε)-approximate solution. Finding Qp Our algorithm is based on the following lemma, whose (simple) proof we defer until the next paragraph: Lemma 1. Qp is flush with P , i.e., one of the edges of Qp contains an edge of P . By Lemma 1, we may consider each edge of P as a candidate flush edge with Qp and turn the scene into simple polygon P (Fig. 1). This reduces finding Qp to solving n instances of the problem of finding a shortest (k + 1)-link path in simple polygon P , a problem which can be solved in polynomial time [12]. Thus we have our main result: Theorem 2. Qp can be found in polynomial time.