On three soft rectangle packing problems with guillotine constraints

TL;DR

This paper studies three guillotine-cutting problems for partitioning rectangles into smaller rectangles with given areas, focusing on minimizing perimeter-related objectives, and provides algorithms and formulations for solving them efficiently or approximately.

Contribution

It introduces optimal and heuristic methods for three rectangle packing problems with guillotine constraints, including polynomial-time solution for one and MIP formulations for NP-hard cases.

Findings

First problem solvable in O(n log n) time

Second and third problems are NP-hard

Proposed MIP and binary search methods effectively solve NP-hard cases

Abstract

We investigate how to partition a rectangular region of length $L_1$ and height $L_2$ into $n$ rectangles of given areas $(a_1, \dots, a_n)$ using two-stage guillotine cuts, so as to minimize either (i) the sum of the perimeters, (ii) the largest perimeter, or (iii) the maximum aspect ratio of the rectangles. These problems play an important role in the ongoing Vietnamese land-allocation reform, as well as in the optimization of matrix multiplication algorithms. We show that the first problem can be solved to optimality in $\mathcal{O}(n \log n)$, while the two others are NP-hard. We propose mixed integer programming (MIP) formulations and a binary search-based approach for solving the NP-hard problems. Experimental analyses are conducted to compare the solution approaches in terms of computational efficiency and solution quality, for different objectives.