Maximum Area Axis-Aligned Square Packings

Given a point set S={s_1,...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 ith 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))>= 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

Flattening Polygonal Linkages via Uniform Angular Motion

We study the motion of polygonal linkages under the restriction that the angles between adjacent edges change uniformly to 0, π, or 2π. We show that convex polygons, orthogonally convex polygons and orthogonal 2-terraines unfold without self-intersection to a straight line in this model, but there exists an orthogonal 12-gon that does not. Further, we show that regular polygons, triangles, quadrilaterals, and convex pentagons can be reconfigured into flat zigzag chains; and every m x n rectangle made of unit-length edges can be reconfigured into a unit-length zigzag