Packing squares independently

TL;DR

This paper introduces the square independent packing problem (SIPP), proposing efficient solution representations, proving NP-hardness, and developing an FPTAS and mathematical programming formulations, validated by computational experiments.

Contribution

It presents a compact solution representation for SIPP, proves its NP-hardness, and offers an FPTAS along with multiple mathematical programming approaches.

Findings

The compact representation reduces search space from Ω(n!) to O(2^n).

The problem is NP-hard.

The proposed algorithms perform well in computational experiments.

Abstract

Given a set of squares and a strip of bounded width and infinite height, we consider a square strip packaging problem, which we call the square independent packing problem (SIPP), to minimize the strip height so that all the squares are packed into independent cells separated by horizontal and vertical partitions. For the SIPP, we first investigate efficient solution representations and propose a compact representation that reduces the search space from $\Omega(n!)$ to $O(2^n)$, with $n$ the number of given squares, while guaranteeing that there exists a solution representation that corresponds to an optimal solution. Based on the solution representation, we show that the problem is NP-hard, and then we propose a fully polynomial-time approximation scheme (FPTAS) to solve it. We also propose three mathematical programming formulations based on different solution representations and confirm the performance of these algorithms through computational experiments. Finally, we discuss several extensions that are relevant to practical applications.