# Packing unequal rectangles and squares in a fixed size circular container using formulation space search

arXiv: https://arxiv.org/abs/1802.07519 · HTML: https://mlsanity.com/paper/1802.07519/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Feb 2018.

## Built on by (most cited first, 2 of 2)

- [CO-Bench: Benchmarking Language Model Agents in Algorithm Search for Combinatorial Optimization](https://mlsanity.com/paper/2504.04310/) (2025)
- [GeoNest: Learning to Select Failure-Aware Neighborhoods for the Irregular Knapsack Problem in a Circular Container](https://mlsanity.com/paper/2609.38863/) (2026)

## Abstract (arXiv)

In this paper we formulate the problem of packing unequal rectangles/squares into a fixed size circular container as a mixed-integer nonlinear program. Here we pack rectangles so as to maximise some objective (e.g. maximise the number of rectangles packed or maximise the total area of the rectangles packed). We show how we can eliminate a nonlinear maximisation term that arises in one of the constraints in our formulation. We indicate the amendments that can be made to the formulation for the special case where we are maximising the number of squares packed. A formulation space search heuristic is presented and computational results given for publicly available test problems involving up to 30 rectangles/squares. Our heuristic deals with the case where the rectangles are of fixed orientation (so cannot be rotated) and with the case where the rectangles can be rotated through ninety degrees.