Dissection with the Fewest Pieces is Hard, Even to Approximate

We prove that it is NP-hard to dissect one simple orthogonal polygon into another using a given number of pieces, as is approximating the fewest pieces to within a factor of $1+1/1080-\varepsilon$.

💡 Research Summary

The paper studies the computational complexity of dissecting one simple orthogonal polygon into another of equal area, focusing on two problems: (i) the decision problem k‑Piece Dissection, which asks whether a given source polygon P can be cut into at most k pieces that can be rearranged (by translation, optionally rotation and reflection) to form a target polygon Q, and (ii) the optimization problem Min Piece Dissection, which seeks the smallest possible number of pieces needed for such a transformation.

The authors prove that both problems are computationally intractable. First, they show that k‑Piece Dissection is NP‑hard by a polynomial‑time reduction from the classic NP‑complete 5‑Partition problem. In a 5‑Partition instance we are given a multiset A of n = 5m integers and must partition it into m groups each summing to the same value p = (∑a_i)/m. The reduction constructs a source polygon P consisting of a long “bar” and a sequence of thin element rectangles, each rectangle having width equal to an integer a_i and height 1, spaced far apart by a distance d_s. The target polygon Q consists of a bar of the same dimensions and a sequence of n/5 “partition rectangles” each of width p and height 1, spaced by a larger distance d_t. The number of allowed pieces k is set to n (the number of integers). By choosing d_s and d_t sufficiently large, any piece that contains parts of two different element rectangles cannot fit inside a single partition rectangle, and no piece can span more than one partition rectangle.

A technical core of the proof is Lemma 1, which states that in any feasible dissection each piece can intersect at most one “trimmed element rectangle” (the upper part of an element rectangle after removing a thin strip of height 4δ). This forces a one‑to‑one correspondence between pieces and element rectangles. Consequently, a feasible dissection yields a packing of the trimmed rectangles into the partition rectangles, which directly translates into a valid 5‑Partition solution. Conversely, a 5‑Partition solution can be turned into a dissection with exactly n pieces by cutting each element rectangle off the bar and placing the pieces into the appropriate partition rectangles. Hence k‑Piece Dissection is NP‑hard; the paper also discusses why membership in NP is unclear because the pieces may need to be described with real coordinates.

The second main result concerns approximation hardness for Min Piece Dissection. The authors first establish a gap‑hardness for Max‑5‑Partition, the optimization version that asks for the maximum number of groups of five elements that sum to p. Using known hardness of approximation for 4‑Uniform 4‑Dimensional Matching (4DM), they prove that distinguishing between instances where the optimum is at least n(1 − ε)/5 and those where it is at most n(1/α + ε)/5 is NP‑hard for some constant α > 1 (they achieve α ≈ 216/215).

They then apply the same geometric reduction as before. If the Max‑5‑Partition instance has a large optimum, the constructed Min Piece Dissection instance can be solved with at most n(1 + ε/5) pieces; if the optimum is small, any solution requires at least α·n(1 − ε/5) pieces. This yields a gap-preserving reduction, implying that approximating Min Piece Dissection within any factor smaller than α ≈ 1 + (1 − 1/α)/5 is NP‑hard. By choosing parameters carefully they obtain the concrete bound 1 + 1/1080 − ε: approximating the minimum number of pieces to within this factor is NP‑hard.

The paper also examines several variants: allowing or forbidding rotations and reflections, requiring pieces to be Jordan regions (connected, simple‑boundary pieces), and restricting the input polygons to be simple orthogonal. The hardness results persist across these variants.

Overall, the work provides the first systematic complexity analysis of polygon dissection with respect to the number of pieces. It shows that even for very restricted families of polygons (simple orthogonal) and very natural operations (translation, optional rotation/reflection), deciding whether a dissection exists with a given number of pieces is NP‑hard, and minimizing the number of pieces cannot be approximated within a modest constant factor unless P = NP. This has implications for fields such as computational geometry, manufacturing (cutting stock problems), puzzle design, and any domain where one wishes to transform shapes with as few cuts as possible.