Packing 2×2 unit squares into grid polygons is NP-complete

In a packing problem, the goal is to put some small objects disjointly into a large container, while optimizing some objective function. The packing problem is very general, and a rich variety of objects and containers are possible. One major split in the taxonomy of packing concerns whether the objects to be packed are allowed to be different or are all identical. In the case of identical packing, probably the simplest non-trivial variant is the following problem, which we call the 2 × 2 packing problem: How many axis-aligned 2× 2 squares can be packed inside a polygon P with n edges drawn on a unit grid, where the squares must be packed such that each occupies exactly four grid locations (i.e. rotation or fractional placement is forbidden)? Whether the problem when P is restricted to be simple (the 2× 2 simple packing problem) is polynomial or NP -complete appears on the open problem project [1] as problem 56. There they cite [3] as proving the 2× 2 packing problem NP complete (where P must be allowed to have holes). This is not true, as in [3] they do prove NP -completeness, but only when all possible locations where the squares could be packed are explicitly provided in the input. Since the size of a normal representation of a grid polygon using binary integers can differ exponentially from the number of possible packing locations inside the polygon (e.g. a k×k square requires Θ(log k) bits to represent but has Θ(k) packing locations), it does not follow from their result that the 2 × 2 packing problem is in NP if the input is simply a polygon. The result of [3] does prove NP -hardness for 2 × 2 packing problem. This paper is devoted to proving the 2 × 2 packing problem is in NP , and thus, combined with the work of [3] proves NP -completeness for the first time. Whether the 2× 2 simple packing problem is polynomial or NP -complete remains an important open question, and remains open quite surprisingly even if P is restricted to be orthogonally convex. A simple greedy algorithm works for Manhattan Skyline polygons [2], the most complex class of polygons for which a polynomial algorithm is known.