Packing squares independently

Wei Wu, Hiroki Numaguchi, Nir Halman, Yannan Hu, Mutsunori Yagiura

Research output: Contribution to journalArticlepeer-review

Abstract

Given a set of squares and a strip with 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 Ω(n!) to O(2n), 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. To solve the SIPP, we propose a dynamic programming method that can be extended to a fully polynomial-time approximation scheme (FPTAS). We also propose three mathematical programming formulations based on different solution representations and confirm their performance through computational experiments with a mathematical programming solver. Finally, we discuss several extensions that are relevant to practical applications.

Original languageEnglish
Article number114910
JournalTheoretical Computer Science
Volume1024
DOIs
StatePublished - 12 Jan 2025

Keywords

  • Complexity
  • Dynamic programming
  • Fully polynomial-time approximation scheme
  • Strip packing

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • General Computer Science

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