Bottom-Left Placement Theorem for Rectangle Packing

This paper proves a bottom-left placement theorem for the rectangle packing problem, stating that if it is possible to orthogonally place n arbitrarily given rectangles into a rectangular container without overlapping, then we can achieve a feasible packing by successively placing a rectangle onto a bottom-left corner in the container. This theorem shows that even for the real-parameter rectangle packing problem, we can solve it after finite times of bottom-left placement actions. Based on this theorem, we might develop efficient heuristic algorithms for solving the rectangle packing problem.

💡 Research Summary

The paper addresses the classic two‑dimensional orthogonal rectangle packing problem (RP): given n rectangles with real‑valued widths wj and heights hj (each may be rotated by 90°) and a rectangular container of width W and height H, decide whether all rectangles can be placed inside the container without overlap. The authors first review existing approaches. The “lattice” methods enumerate all integer‑grid positions and are exact only for integer‑parameter instances, while most practical algorithms rely on placement heuristics that restrict candidate positions to a small set. A well‑known heuristic, the Bottom‑Left (BL) rule (place each rectangle as low and as far left as possible), is known to be incomplete: there exist feasible packings that cannot be obtained by any ordering of rectangles under the BL rule.

The central contribution is the Bottom‑Left Placement Theorem, which proves that for the general RP problem with real parameters, a feasible packing can always be constructed by a sequence of “bottom‑left placement actions”. A bottom‑left placement action places a rectangle onto a bottom‑left corner formed by already placed rectangles and the container walls, and then the rectangle becomes bottom‑left stable (it cannot move further down or left without overlapping another rectangle). The theorem consists of three main logical steps:

1. Lemma 1 (Existence of a bottom‑left stable packing).
   Starting from any feasible (non‑overlapping) packing X₀, the authors define a continuous overlap‑area function O(X) over ℝ^{2n} and consider the zero‑set S₀ = {X | O(X)=0}. S₀ is non‑empty, closed, and bounded. They then minimize the linear functional L = Σ_i (x_i + y_i) over S₀. At a minimizer X*, no rectangle can be shifted left or down without creating overlap, otherwise a point with a smaller L would exist, contradicting optimality. Hence X* is a feasible packing where every rectangle is bottom‑left stable. This argument provides a rigorous proof that any feasible packing can be transformed into a bottom‑left stable one without changing orientations.

2. Lemma 2 (Escaping Lemma).
   In any feasible packing, after removing the four container borders, there always exists at least one rectangle that can move freely upward and rightward. The proof orders the top‑right corners of all rectangles lexicographically by (x, y) and iteratively selects the rightmost rectangle among those that lie above the current candidate. Because the set is finite, the process terminates with a rectangle that has no other rectangle above it and none to its right, guaranteeing free upward‑rightward motion.

3. Theorem 1 (Construction of a bottom‑left insertion order).
   Using Lemma 2, the authors repeatedly remove a rectangle that can move freely upward and rightward, thereby generating a removal sequence of length n. Reversing this sequence yields an insertion order. For each i (1 ≤ i ≤ n) in this order, the i‑th rectangle can be placed onto a bottom‑left corner formed by rectangles 1…i‑1 and the container walls. Consequently, any bottom‑left stable packing can be reproduced by a series of bottom‑left placement actions.

4. Theorem 2 (Bottom‑Left Placement Theorem).
   Combining Lemma 1 (any feasible packing can be turned into a bottom‑left stable one) with Theorem 1 (any bottom‑left stable packing can be generated by bottom‑left placements) yields the main result: if a feasible packing exists for the given set of rectangles, then there exists a sequence of bottom‑left placement actions that constructs a feasible packing. This establishes the completeness of the bottom‑left placement strategy for the general RP problem with real parameters.

5. Theorem 3 (Finite termination).
   The authors bound the total number of possible bottom‑left placement actions by n! · 2ⁿ · θⁿ, where n! accounts for rectangle permutations, 2ⁿ for orientation choices, and θⁿ for the number of possible bottom‑left corners per rectangle. Since this bound is finite, an algorithm that systematically explores all bottom‑left placement actions will terminate after a finite number of arithmetic and logical operations, either finding a feasible packing or correctly reporting infeasibility.

The paper concludes by noting that the theorem provides a solid theoretical foundation for designing efficient exact and heuristic algorithms for rectangle packing. The authors suggest future work on extending the concept to three‑dimensional orthogonal packing (box packing) and on developing practical heuristics that exploit the bottom‑left corner structure. Overall, the work bridges a long‑standing gap: it proves that a simple, geometrically intuitive placement rule—placing each rectangle onto a bottom‑left corner—suffices to solve the general 2‑D rectangle packing problem, thereby offering both theoretical insight and a practical pathway for algorithmic development.