fastxgemm-n2r7s4t1

decomposition benchmark_suitable variable_bound set_partitioning set_covering mixed_binary general_linear

Submitter	Variables	Constraints	Density	Status	Group	Objective	MPS File
Laurent Sorber	904	6972	3.58323e-03	easy	fastxgemm	41.99999999999998	fastxgemm-n2r7s4t1.mps.gz

Naive multiplication of two N by N matrices requires N^3 scalar multiplications. For N=2, Strassen showed that it could be done in only R=7 < 8=N^3 multiplications. For N=3, it is known that 19 <= R <= 23, and for N=4 it is known that 34 <= R <= 49. This repository contains code that generates a mixed-integer linear program (MILP) formulation of the fast matrix multiplication problem for finding solutions with R < N^3 and proving that they are optimal. For a more detailed description, see the accompanying manuscript.

Location of instance in collection

Instance Statistics

Detailed explanation of the following tables can be found here.

Size Related Properties
	Original	Presolved
Variables	904	904
Constraints	6972	6972
Binaries	56	56
Integers	0	168
Continuous	848	680
Implicit Integers	0	168
Fixed Variables	0	0
Nonzero Density	0.00358323	0.00358323
Nonzeroes	22584	22584

Constraint Classification Properties
	Original	Presolved
Total	6972	6972
Empty	0	0
Free	0	0
Singleton	0	0
Aggregations	0	0
Precedence	1428	0
Variable Bound	1428	2856
Set Partitioning	0	84
Set Packing	0	0
Set Covering	0	33
Cardinality	0	0
Invariant Knapsack	0	0
Equation Knapsack	0	0
Bin Packing	0	0
Knapsack	0	0
Integer Knapsack	0	0
Mixed Binary	4116	331
General Linear	0	3668
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	0.903090
Constraint %		13.2817	13.2817	13.2817	13.2817
Variable %		11.0619	11.0619	11.0619	11.0619
Score	0.826874

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	42	42	0	0	0	-	2018-10-13	Solution found during MIPLIB2017 problem selection.

Similar instances in collection

The following instances are most similar to fastxgemm-n2r7s4t1 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	Continuous	Constraints	Nonz.	Submitter	Group	Objective	Tags
fastxgemm-n2r6s0t2	easy	784	48	0	736	5998	19376	Laurent Sorber	fastxgemm	230	benchmark decomposition benchmark_suitable variable_bound set_partitioning set_covering mixed_binary general_linear
fastxgemm-n3r21s3t6	open	18684	378	0	18306	219368	718146	Laurent Sorber	fastxgemm	4110.99800117485*	decomposition variable_bound set_partitioning set_covering mixed_binary general_linear
fastxgemm-n3r22s4t6	open	19539	396	0	19143	229742	752274	Laurent Sorber	fastxgemm	3101.998498499999*	decomposition variable_bound set_partitioning set_covering mixed_binary general_linear
fastxgemm-n3r23s5t6	open	20394	414	0	19980	240116	786402	Laurent Sorber	fastxgemm	3089.997998*	decomposition variable_bound set_partitioning set_covering mixed_binary general_linear
neos-4335793-snake	hard	30827	20473	7865	2489	37166	129119	Jeff Linderoth	neos-pseudoapplication-44	27	numerics aggregations precedence variable_bound set_packing cardinality invariant_knapsack knapsack mixed_binary general_linear

Reference

@misc{Sorber2017,
author = {Laurent Sorber and Marc Van Barel},
title = {{A mixed-integer linear program formulation for fast matrix multiplication}},
howpublished = "\url{https://github.com/lsorber/fast-matrix-multiplication/blob/master/latex/fast-matrix-multiplication.pdf}",
day = {30},
month = {April},
year = {2017}, 
note = "[Online]"
}