Algorithms for Fair Partitioning of Convex Polygons

A survey of mass partitions

“…Even though Theorem 3.0.1 deals with fair partitions of measures, the proofs of Karasev, Hubard, and Aronov, and of Blagojević and Ziegler yield much more [KHA14,BZ14]. They were motivated by a question of Nandakumar and Ramana Rao, which asked if every polygon in the plane could be split into k convex parts of equal area and equal perimeter, for any positive integer k. If the polygon has n vertices, there are algorithms that find such a partition for k = 2 h in O((2n) h ) time [AD15].…”

Cutting Polygons into Small Pieces with Chords: Laser-Based Localization

“…In these problems, the partition is not restricted to chords (or hyperplanes). Topological methods are used for existential results in this area, and very few algorithmic results are known [1]. Another related problem is the family of so-called cake cutting problems [15,31], in which an infinite straight line "knife" is used to cut a convex "cake" into (convex) pieces that represent a "fair" division into portions.…”

“…, n, if τ (x i ) = true, then place a horizontal laser at y = 3(i − 1) + 1 (along the bottom corridor touching room for x i ), otherwise at y = 3(i − 1) + 2 (along the top corridor touching room for x i ). These lasers split each variable room into two rectangles of area 1 2 and 3 2 . For j = 1, .…”