square31

benchmark_suitable set_partitioning general_linear

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.)

Location of instance in collection

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.

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 %		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	Status	Variables	Binaries	Integers	Constraints	Nonz.	Submitter	Group	Objective	Tags
square37	easy	49320	49284	36	33150	9475672	Sascha Kurz	square	14.9999997973	benchmark_suitable set_partitioning general_linear
square23	easy	11660	11638	22	7887	898813	Sascha Kurz	square	13	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
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},
}