Circle Covering Representation for Nesting problems with continuous rotations

Heaviness in Circle Rotations

We are concerned with describing the structure of the set of points in the unit interval which, when subjected to rotation by irrational α modulo one, for all finite portions of the orbit contain at least as many points in the bottom half of the interval as in the top half. Specifically, an inductive procedure for describing the set based on the continued fraction expansion of α is developed, l...

Covering on a Circle

Dynamical circle covering with homogeneous Poisson updating

We consider a dynamical variant of Dvoretzky’s classical covering problem of the unit circumference circle, where the centers of the arcs are updated according to independent Poisson processes of unit intensity. This dynamical model was introduced (in greater generality) in [5], where is was shown that when the length of the n’th arc is `n and we write un := ∏n k=1(1 − `k), then lim infn n(log ...

A cross entropy algorithm for continuous covering location problem

Covering problem tries to locate the least number of facilities and each demand has at least one facility located within a specific distance.This paper considers a cross entropy algorithm for solving the mixed integer nonlinear programming (MINLP) for covering location model.The model is solved to determine the best covering value.Also, this paper proposes aCross Entropy (CE) algorithm consider...

Edge-Matching Problems with Rotations

Edge-matching problems, also called edge matching puzzles, are abstractions of placement problems with neighborhood conditions. Pieces with colored edges have to be placed on a board such that adjacent edges have the same color. The problem has gained interest recently with the (now terminated) Eternity II puzzle, and new complexity results. In this paper we consider a number of settings which ...