NP-Hard Triangle Packing Problems

In computational geometry, packing problems ask whether a set of rigid pieces can be placed inside a target region such that no two pieces overlap. The triangle packing problem is a packing problem that involves triangular pieces, and it is crucial for algorithm design in many areas, including origami design, cutting industries, and warehousing. Previous works in packing algorithms have conjectured that triangle packing is NP-hard. In this paper, we mathematically prove this conjecture. We prove the NP-hardness of three distinct triangle packing problems: (i) packing right triangles into a rectangle, (ii) packing right triangles into a right triangle, and (iii) packing equilateral triangles into an equilateral triangle. We construct novel reductions from the known NP-complete problems 3-partition and 4-partition. Furthermore, we generalize that packing arbitrary triangles into an arbitrary target region is strongly NP-hard. Because triangle packing is NP-hard, triangle packing must be determined by approximation or heuristic algorithms rather than exact algorithms.