Submitter Variables Constraints Density Status Group Objective MPS File
Sascha Kurz 28860 19435 7.01951e-03 easy square 15 square31.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 28860 10445
Constraints 19435 1020
Binaries 28830 10415
Integers 30 30
Continuous 0 0
Implicit Integers 0 0
Fixed Variables 0 0
Nonzero Density 0.00701951 0.10594700
Nonzeroes 3937200 1128750
Constraint Classification Properties
Original Presolved
Total 19435 1020
Empty 0 0
Free 0 0
Singleton 18415 0
Aggregations 0 0
Precedence 0 0
Variable Bound 0 0
Set Partitioning 961 961
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 59 59
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 % 94.2157 94.2157 94.2157 94.2157
Variable % 99.7128 99.7128 99.7128 99.7128
Score 0.002706

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 square31 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
square37 49320 49284 36 0 33150 9475670 Sascha Kurz square easy 15.0000
square23 11660 11638 22 0 7887 898813 Sascha Kurz square easy 13.0000
square41 62234 62197 37 0 40160 13566400 Sascha Kurz square easy 15.0000
square47 95030 94987 43 0 61591 27329900 Sascha Kurz square easy 16.0000
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 19, 2018 by Gregor Hendel
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