Peeling Potatoes Near-Optimally in Near-Linear Time

Abstract

We consider the following geometric optimization problem: find a convex polygon of maximum area contained in a given simple polygon P with n vertices. We give a randomized near-linear-time (1 − ε)-approximation algorithm for this problem: in O(n(log 2 n + (1/ε 3 ) log n + 1/ε 4 )) time we find a convex polygon contained in P that, with probability at least 2/3, has area at least (1 − ε) times the area of an optimal solution. We also obtain similar results for the variant of computing a convex polygon inside P with maximum perimeter.To achieve these results we provide new results in geometric probability. The first result is a bound relating the probability that two points chosen uniformly at random inside P are mutually visible and the area of the largest convex body inside P . The second result is a bound on the expected value of the difference between the perimeter of any planar convex body K and the perimeter of the convex hull of a uniform random sample inside K.

On the Beer Index of Convexity and Its Variants

“…Let B be a point in V 1 (A). Using a similar approach to the one used by Cabello et al [6] in the proof of Theorem 1.1, we show that…”

“…Clearly, the set T(A) is measurable. Summing the three estimates on areas of the trapezoids, we obtain The following lemma is a slightly more general version of a result of Cabello et al [6].…”

“…We will show that each p-component of L i ∪ L i+1 is a rooted set, with the extra property that all its points are reachable from its root by a polygonal line with at most two segments (Lemma 2.5). To handle such sets, we will generalize the techniques that Cabello et al [6] have used to handle weakly star-shaped sets in their proof of Theorem 1.1. We will assign to every point A ∈ R a set T(A) of measure O(smc(R)), such that for every (A, B) ∈ Seg(R), we have either B ∈ T(A) or A ∈ T(B) (Lemma 2.7).…”

“…For polygons that are not weakly star-shaped, Cabello et al [6] gave a superlinear bound. Moreover, Cabello et al [6] conjectured that even for a general simple polygon P , b(P ) can be bounded from above by a linear function of c(P ). (The question whether b(P ) = O(c(P )) for simple polygons P was originally asked by Cabello and Saumell [7].)…”

CofiFab

“…Our problem is close to the potato peeling [Goodman 1981], which finds a convex polygon/polyhedron of maximum size in a given simple 2D/3D shape. Only a few works have addressed this NP-hard problem [Borgefors and Strand 2005] [Deits and Tedrake 2015], mostly in 2D [Chang and Yap 1986] [Cabello et al 2014]. In sharp contrast, we employ multiple polyhedrons to approximate the object interior and enforce a number of geometric constraints for fabrication and assembly.…”