Exact algorithms for a geometric packing problem

Abstract

We investigate the following packing problem: given n points Pi,…,Pn in the plane determine the supremum of all reals σ, for which there are n pairwise disjoint, axis-parallel squares Qi,…, Qn of side length σ, where for each i, 1 ≤ t ≤ n, p; is a corner of Qi. The problem arises in the connection with lettering of maps, and its decision version is NP-complete. We present two exact algorithms for the decision problem with time complexities 4O(n) and 4O(n log n), resp. While the first one is of only theoretical interest because of a large multiplicative factor in the exponent, the other is suitable for practical computation.