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 42 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.

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.

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 21084* decomposition variable_bound set_partitioning set_covering mixed_binary general_linear
fastxgemm-n3r22s4t6 open 19539 396 0 19143 229742 752274 Laurent Sorber fastxgemm 21084* decomposition variable_bound set_partitioning set_covering mixed_binary general_linear
fastxgemm-n3r23s5t6 open 20394 414 0 19980 240116 786402 Laurent Sorber fastxgemm 27087* decomposition variable_bound set_partitioning set_covering mixed_binary general_linear
neos-4335793-snake open 30827 20473 7865 2489 37166 129119 Jeff Linderoth neos-pseudoapplication-44 43.00000000009* 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]"
}

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