Maximum Area Axis-Aligned Square Packings

Abstract

Given a point set $S=\{s_1,\dots ,s_n\}$ in the unit square $U=[0,1{]}^2$, an anchored square packing is a set of $n$ interior-disjoint empty squares in $U$ such that $s_i$ is a corner of the $i$th square. The reach $R(S)$ of $S$ is the set of points that may be covered by such a packing, that is, the union of all empty squares anchored at points in $S$. It is shown that area$(R(S))\ge \frac{1}{2}$ for every finite set $S\subset U$, and this bound is the best possible. The region $R(S)$ can be computed in $O(n\log n)$ time. Finally, we prove that finding a maximum area anchored square packing is NP-complete. This is the first hardness proof for a geometric packing problem where the size of geometric objects in the packing is unrestricted.

20 pages, 13 figures. A 15-page extended abstract appears in the Proceedings of the 43rd International Symposium on Mathematical Foundations of Computer Science (Liverpool, UK, 2018)