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.)

Instance Statistics

Detailed explanation of the following tables can be found here.

Size Related Properties
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
Constraint Classification Properties
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

Structure

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

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

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

Similar instances in collection

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 Variables Binaries Integers Continuous Constraints Nonz. Submitter Group Status Objective
square47 95030 94987 43 0 61591 27329900 Sascha Kurz square easy 16.0000
square37 49320 49284 36 0 33150 9475670 Sascha Kurz square easy 15.0000
square31 28860 28830 30 0 19435 3937200 Sascha Kurz square easy 15.0000
square23 11660 11638 22 0 7887 898813 Sascha Kurz square easy 13.0000
ivu52 157591 157591 0 0 2116 2179480 S. Weider ivu hard 481.0068

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 09, 2018 by Gregor Hendel
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