Placing text boxes on graphs. A fast approximation algorithm for maximizing overlap of a square and a simple polygon

Abstract

In this paper we consider the problem of placing a unit square on a face of a drawn graph bounded by n vertices such that the area of overlap is maximized. Exact algorithms are known that solve this problem in O(n 2) time. We present an approximation algorithm that-for any given ε>∈0- places a (1∈+∈ε)-square on the face such that the area of overlap is at least the area of overlap of a unit square in an optimal placement. The algorithm runs in time. Extensions of the algorithm solve the problem for unit discs, using time, and for bounded aspect ratio rectangles of unit area, using time. © 2009 Springer Berlin Heidelberg.