Submitter Variables Constraints Density Status Group Objective MPS File
Sascha Kurz 11660 7887 9.7737e-03 easy square 13 square23.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.)

Instance Statistics

Detailed explanation of the following tables can be found here.

Size Related Properties
Original Presolved
Variables 11660 4345
Constraints 7887 572
Binaries 11638 4323
Integers 22 22
Continuous 0 0
Implicit Integers 0 0
Fixed Variables 0 0
Nonzero Density 0.0097737 0.1084950
Nonzeroes 898813 269646
Constraint Classification Properties
Original Presolved
Total 7887 572
Empty 0 0
Free 0 0
Singleton 7315 0
Aggregations 0 0
Precedence 0 0
Variable Bound 0 0
Set Partitioning 529 529
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 43 43
Indicator 0 0

Structure

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

value min median mean max
Components 0.301030
Constraint % 92.4825 92.4825 92.4825 92.4825
Variable % 99.4937 99.4937 99.4937 99.4937
Score 0.004683

Best Known Solution(s)

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

Similar instances in collection

The following instances are most similar to square23 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 Variables Binaries Integers Continuous Constraints Nonz. Submitter Group Status Objective
square31 28860 28830 30 0 19435 3937200 Sascha Kurz square easy 15
square37 49320 49284 36 0 33150 9475670 Sascha Kurz square easy 15
square41 62234 62197 37 0 40160 13566400 Sascha Kurz square easy 15
square47 95030 94987 43 0 61591 27329900 Sascha Kurz square easy 16
mod010 2655 2655 0 0 146 11203 MIPLIB submission pool mod easy 6548

Reference

@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},
}

Last Update Nov 19, 2018 by Gregor Hendel
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