Submitter | Variables | Constraints | Density | Status | Group | Objective | MPS File |
---|---|---|---|---|---|---|---|

Sascha Kurz | 62234 | 40160 | 5.42805e-03 | easy | square | 15 | square41.mps.gz |

Squaring the square For a given integer n, determine the minimum number of squares in a tiling of an \(n\times n\) square using using only integer sided squares of smaller size. (Although the models get quite large even for moderate n, they can be solved to optimality for all \(n \le 61\), while challenging the MIP solver, especially the presolver.)

Detailed explanation of the following tables can be found here.

Original | Presolved | |
---|---|---|

Variables | 62234 | 23828 |

Constraints | 40160 | 1754 |

Binaries | 62197 | 23791 |

Integers | 37 | 37 |

Continuous | 0 | 0 |

Implicit Integers | 0 | 0 |

Fixed Variables | 0 | 0 |

Nonzero Density | 0.00542805 | 0.10375900 |

Nonzeroes | 13566400 | 4336550 |

Original | Presolved | |
---|---|---|

Total | 40160 | 1754 |

Empty | 0 | 0 |

Free | 0 | 0 |

Singleton | 38406 | 0 |

Aggregations | 0 | 0 |

Precedence | 0 | 0 |

Variable Bound | 0 | 0 |

Set Partitioning | 1681 | 1681 |

Set Packing | 0 | 0 |

Set Covering | 0 | 0 |

Cardinality | 0 | 0 |

Invariant Knapsack | 0 | 0 |

Equation Knapsack | 0 | 0 |

Bin Packing | 0 | 0 |

Knapsack | 0 | 0 |

Integer Knapsack | 0 | 0 |

Mixed Binary | 0 | 0 |

General Linear | 73 | 73 |

Indicator | 0 | 0 |

Available nonzero structure and decomposition information. Further information can be found here.

Decomposed structure of original problem (dec-file)

Decomposed structure after trivial presolving (dec-file)

value | min | median | mean | max | |
---|---|---|---|---|---|

Components | 0.301030 | ||||

Constraint % | 95.8381 | 95.8381 | 95.8381 | 95.8381 | |

Variable % | 99.8447 | 99.8447 | 99.8447 | 99.8447 | |

Score | 0.001488 |

Find solutions below. Download the archive containing all solutions from the Download page.

ID | Objective | Exact | Int. Viol | Cons. Viol | Obj. Viol | Submitter | Date | Description |
---|---|---|---|---|---|---|---|---|

1 | 15 | 15 | 0 | 0 | 0 | - | 2018-10-13 | Solution found during MIPLIB2017 problem selection. |

The following instances are most similar to square41 in the collection. This similarity analysis is based on 100 scaled instance features describing properties of the variables, objective function, bounds, constraints, and right hand sides.

Instance | Status | Variables | Binaries | Integers | Continuous | Constraints | Nonz. | Submitter | Group | Objective | Tags |
---|---|---|---|---|---|---|---|---|---|---|---|

square47 | easy | 95030 | 94987 | 43 | 0 | 61591 | 27329856 | Sascha Kurz | square | 15.9999999997877 | benchmark benchmark_suitable set_partitioning general_linear |

square37 | easy | 49320 | 49284 | 36 | 0 | 33150 | 9475672 | Sascha Kurz | square | 14.9999997973 | benchmark_suitable set_partitioning general_linear |

square31 | easy | 28860 | 28830 | 30 | 0 | 19435 | 3937200 | Sascha Kurz | square | 15 | benchmark_suitable set_partitioning general_linear |

square23 | easy | 11660 | 11638 | 22 | 0 | 7887 | 898813 | Sascha Kurz | square | 13 | benchmark_suitable set_partitioning general_linear |

ivu52 | hard | 157591 | 157591 | 0 | 0 | 2116 | 2179476 | S. Weider | ivu | 481.0068 | binary set_partitioning invariant_knapsack knapsack mixed_binary |

```
@article{kurz2012squaring,
title={Squaring the square with integer linear programming},
author={Kurz, Sascha},
journal={Journal of Information Processing},
volume={20},
number={3},
pages={680--685},
year={2012},
}
```

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