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Framework for $\exists \mathbb{R}$-Completeness of Two-Dimensional Packing Problems

Mikkel Abrahamsen, Tillmann Miltzow, Nadja Seiferth

TL;DR

This work establishes a general framework to prove $\exists \mathbb{R}$-hardness for a broad class of two-dimensional packing problems by encoding existential real equations as geometric placement constraints. It introduces a reduction pipeline from Curve-ETR$[f,g]$ via wired wiring diagrams to packing instances, employing a suite of gadgets (Anchor, Swap, Split, Adder, Swing, Gramophone) and a fingerprinting technique that leverages minute slack to force near-canonical placements and consistent variable encoding. The authors prove strong hardness results, showing that many variants, including packing convex/curved pieces into polygonal or square containers with translations or rigid motions, are $\exists \mathbb{R}$-hard, and they extend the reductions to a square container via a 4-monotone intermediate container. These results imply that widely used ILP/SAT-based heuristics may be unsuitable for many rotations and non-polygonal-shape packing problems, highlighting the need for new algorithmic approaches when nonlinearity and geometric complexity are present. The framework also demonstrates strong hardness with rational-coordinate gadgets, suggesting robustness and potential applicability to other geometric-computation problems beyond packing.

Abstract

The aim in packing problems is to decide if a given set of pieces can be placed inside a given container. A packing problem is defined by the types of pieces and containers to be handled, and the motions that are allowed to move the pieces. The pieces must be placed so that in the resulting placement, they are pairwise interior-disjoint. We establish a framework which enables us to show that for many combinations of allowed pieces, containers and motions, the resulting problem is $\exists \mathbb{R}$-complete. This means that the problem is equivalent (under polynomial time reductions) to deciding whether a given system of polynomial equations and inequalities with integer coefficients has a real solution. We consider packing problems where only translations are allowed as the motions, and problems where arbitrary rigid motions are allowed, i.e., both translations and rotations. When rotations are allowed, we show that it is an $\exists \mathbb{R}$-complete problem to decide if a set of convex polygons, each of which has at most $7$ corners, can be packed into a square. Restricted to translations, we show that the following problems are $\exists \mathbb{R}$-complete: (i) pieces bounded by segments and hyperbolic curves to be packed in a square, and (ii) convex polygons to be packed in a container bounded by segments and hyperbolic curves.

Framework for $\exists \mathbb{R}$-Completeness of Two-Dimensional Packing Problems

TL;DR

This work establishes a general framework to prove -hardness for a broad class of two-dimensional packing problems by encoding existential real equations as geometric placement constraints. It introduces a reduction pipeline from Curve-ETR via wired wiring diagrams to packing instances, employing a suite of gadgets (Anchor, Swap, Split, Adder, Swing, Gramophone) and a fingerprinting technique that leverages minute slack to force near-canonical placements and consistent variable encoding. The authors prove strong hardness results, showing that many variants, including packing convex/curved pieces into polygonal or square containers with translations or rigid motions, are -hard, and they extend the reductions to a square container via a 4-monotone intermediate container. These results imply that widely used ILP/SAT-based heuristics may be unsuitable for many rotations and non-polygonal-shape packing problems, highlighting the need for new algorithmic approaches when nonlinearity and geometric complexity are present. The framework also demonstrates strong hardness with rational-coordinate gadgets, suggesting robustness and potential applicability to other geometric-computation problems beyond packing.

Abstract

The aim in packing problems is to decide if a given set of pieces can be placed inside a given container. A packing problem is defined by the types of pieces and containers to be handled, and the motions that are allowed to move the pieces. The pieces must be placed so that in the resulting placement, they are pairwise interior-disjoint. We establish a framework which enables us to show that for many combinations of allowed pieces, containers and motions, the resulting problem is -complete. This means that the problem is equivalent (under polynomial time reductions) to deciding whether a given system of polynomial equations and inequalities with integer coefficients has a real solution. We consider packing problems where only translations are allowed as the motions, and problems where arbitrary rigid motions are allowed, i.e., both translations and rotations. When rotations are allowed, we show that it is an -complete problem to decide if a set of convex polygons, each of which has at most corners, can be packed into a square. Restricted to translations, we show that the following problems are -complete: (i) pieces bounded by segments and hyperbolic curves to be packed in a square, and (ii) convex polygons to be packed in a container bounded by segments and hyperbolic curves.

Paper Structure

This paper contains 22 sections, 33 theorems, 20 equations, 46 figures, 1 table.

Table of Contents

  1. Introduction
  2. Reduction skeleton
  3. The problem Wired-Curve-ETR[f,g]
  4. Constructing a packing instance
  5. Basic tools: Slack, fingerprinting and unique angles
  6. Proof structure of Lemma \ref{['lem:validAligned']}
  7. Proof structure of Lemma \ref{['lem:consistency']}
  8. Square container
  9. Fingerprinting
  10. Fingerprinting a single piece
  11. Fingerprinting more pieces at once
  12. Proof of Single fingerprint (Lemma \ref{['lemma:boundDiff0']})
  13. Generalization to curved polygons
  14. Linear gadgets
  15. Anchor
  16. ...and 7 more sections

Key Result

Theorem 1.1

For any decision problem $P$, there is a real verification algorithm for $P$ if and only if $P \in \exists \mathbb{R}\xspace$.

Figures (46)

  • Figure 4: Real examples of nesting on a leather hide (left) and a piece of fabric (right), kindly provided by MIRISYS and produced by their software for automatic nesting, https://www.mirisys.com/.
  • Figure 5: Left: The optimal packing of five unit squares already requires rotations. Right: The currently best known packing of eleven unit squares into a larger square Gensane2005.
  • Figure 6: A wiring diagram corresponding to the Curve-ETR$[f,g]$ formula $x_2+x_3=x_1\land f(x_2,x_1)\geq 0\land g(x_2,x_1)\geq 0$.
  • Figure 7: A sketch of the instance of Pack$[ \xspace \rightarrow \xspace , ]$ we get from the wiring diagram in \ref{['fig:WiringDiagram']}, broken over six lines. The adders and curvers (swings) are marked with gray boxes. The (light and dark) red, blue and green pieces are the variable pieces and the pieces of each nuance form a lane.
  • Figure 8: A wire $\overrightarrow x$ and a variable piece representing $x$ placed on top, showing the leftmost and rightmost canonical placements of the piece. The large arrow in the piece indicates that the piece is right-oriented.
  • ...and 41 more figures

Theorems & Definitions (39)

  • Theorem 1.1: Erickson, Hoog, Miltzow RobustComputation
  • definition 1.2: Curve-ETR$[f,g]$ formula
  • definition 1.3: Well-behaved and convexly/concavely curved function
  • Theorem 1.4: Miltzow and Schmiermann miltzow2021classifying
  • definition 2.1
  • Lemma 2.2
  • Theorem 2.3
  • corollary 2.4
  • Lemma 2.5: Solution preservation
  • Lemma 2.6: Soundness
  • ...and 29 more