square41

benchmark benchmark_suitable set_partitioning general_linear

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.)

Location of instance in collection

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.

Nonzero entries of original problem

Nonzero entries after trivial presolving

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

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	Status	Variables	Binaries	Integers	Constraints	Nonz.	Submitter	Group	Objective	Tags
square47	easy	95030	94987	43	61591	27329856	Sascha Kurz	square	15.9999999997877	benchmark benchmark_suitable set_partitioning general_linear
square37	easy	49320	49284	36	33150	9475672	Sascha Kurz	square	14.9999997973	benchmark_suitable set_partitioning general_linear
square31	easy	28860	28830	30	19435	3937200	Sascha Kurz	square	15	benchmark_suitable set_partitioning general_linear
square23	easy	11660	11638	22	7887	898813	Sascha Kurz	square	13	benchmark_suitable set_partitioning general_linear
ivu52	hard	157591	157591	0	2116	2179476	S. Weider	ivu	481.0068	binary set_partitioning invariant_knapsack knapsack mixed_binary

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},
}