Corner Occupying Theorem for the Two-dimensional Integral Rectangle Packing Problem

TL;DR

This paper proves a corner occupying theorem for 2D integral rectangle packing, showing that feasible packings can be constructed by successively placing rectangles in bottom-left corners, which could lead to efficient heuristics.

Contribution

It introduces a formal proof of the corner occupying theorem for integral rectangle packing, providing a theoretical foundation for heuristic algorithm development.

Findings

Theorem confirms feasible packings can be built step-by-step from bottom-left corners.

Supports development of efficient heuristic algorithms for rectangle packing.

The theorem has been implicitly recognized but not formally proven before.

Abstract

This paper proves a corner occupying theorem for the two-dimensional integral rectangle packing problem, stating that if it is possible to orthogonally place n arbitrarily given integral rectangles into an integral rectangular container without overlapping, then we can achieve a feasible packing by successively placing an integral rectangle onto a bottom-left corner in the container. Based on this theorem, we might develop efficient heuristic algorithms for solving the integral rectangle packing problem. In fact, as a vague conjecture, this theorem has been implicitly mentioned with different appearances by many people for a long time.