Bottom-Left Placement Theorem for Rectangle Packing

TL;DR

This paper proves a bottom-left placement theorem for rectangle packing, showing that feasible packings can be achieved through successive bottom-left placements, which can inform efficient heuristic algorithms.

Contribution

The paper introduces a bottom-left placement theorem for rectangle packing, enabling finite-step solutions and potential development of efficient heuristics.

Findings

Feasible packings can be constructed via bottom-left placements.

The theorem applies to real-parameter rectangle packing problems.

It provides a foundation for heuristic algorithm development.

Abstract

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.