square37

benchmark_suitable set_partitioning general_linear

Submitter	Variables	Constraints	Density	Status	Group	Objective	MPS File
Sascha Kurz	49320	33150	5.79567e-03	easy	square	14.9999997973	square37.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	49320	17610
Constraints	33150	1440
Binaries	49284	17574
Integers	36	36
Continuous	0	0
Implicit Integers	0	0
Fixed Variables	0	0
Nonzero Density	0.00579567	0.10484000
Nonzeroes	9475670	2658570

Constraint Classification Properties
	Original	Presolved
Total	33150	1440
Empty	0	0
Free	0	0
Singleton	31710	0
Aggregations	0	0
Precedence	0	0
Variable Bound	0	0
Set Partitioning	1369	1369
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	71	71
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 %		95.0694	95.0694	95.0694	95.0694
Variable %		99.7956	99.7956	99.7956	99.7956
Score	0.001943

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	1e-07	0	-	2018-10-13	Solution found during MIPLIB2017 problem selection.

Similar instances in collection

The following instances are most similar to square37 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
square31	easy	28860	28830	30	19435	3937200	Sascha Kurz	square	15	benchmark_suitable set_partitioning general_linear
square41	easy	62234	62197	37	40160	13566426	Sascha Kurz	square	15	benchmark benchmark_suitable set_partitioning general_linear
square47	easy	95030	94987	43	61591	27329856	Sascha Kurz	square	15.9999999997877	benchmark 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},
}