Exact algorithms for a geometric packing problem (extended abstract)

Abstract

We investigate the following packing problem: given n points p ₁,..., p _n in the plane determine the supremum σ _opt of all reals σ, for which there are n pairwise disjoint, axis-parallel squares Q ₁,..., Q _n of side length σ, where for each i, 1 ≤ i ≤ n, p _i is a corner of Q _i . 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 4^O(√n) and 4^{O(√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.