The kissing problem: How to end a gathering when everyone kisses everyone else goodbye

Abstract

This paper introduces the kissing problem: given a rectangular room with n people in it, what is the most efficient way for each pair of people to kiss each other goodbye? The room is viewed as a set of pixels that form a subset of the integer grid. At most one person can stand on a pixel at once, and people move horizontally or vertically. In order to move into a pixel in time step t, the pixel must be empty in time step t-1. The paper gives one algorithm for kissing everyone goodbye. (1) This algorithm is a 4 + o(1)-approximation algorithm in a crowded room (e.g., only one unoccupied pixel). (2) It is a 10 + o(1)-approximation algorithm for kissing in a comfortable room (e.g., at most half the pixels are empty). (3) It is a 25+o(1)-approximation for kissing in a sparse room. © 2012 Springer-Verlag Berlin Heidelberg.