fastxgemm-n3r21s3t6

decomposition variable_bound set_partitioning set_covering mixed_binary general_linear

Submitter Variables Constraints Density Status Group Objective MPS File
Laurent Sorber 18684 219368 1.75214e-04 open fastxgemm 21084* fastxgemm-n3r21s3t6.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 18684 18684
Constraints 219368 219368
Binaries 378 378
Integers 0 1134
Continuous 18306 17172
Implicit Integers 0 1134
Fixed Variables 0 0
Nonzero Density 0.000175214 0.000175214
Nonzeroes 718146 718146
Constraint Classification Properties
Original Presolved
Total 219368 219368
Empty 0 0
Free 0 0
Singleton 0 0
Aggregations 0 0
Precedence 46494 0
Variable Bound 46494 92988
Set Partitioning 0 567
Set Packing 0 0
Set Covering 0 90
Cardinality 0 0
Invariant Knapsack 0 0
Equation Knapsack 0 0
Bin Packing 0 0
Knapsack 0 0
Integer Knapsack 0 0
Mixed Binary 126380 2684
General Linear 0 123039
Indicator 0 0

Structure

Available nonzero structure and decomposition information. Further information can be found here.

value min median mean max
Components 1.342423
Constraint % 4.67707 4.67707 4.67707 4.67707
Variable % 4.33526 4.33526 4.33526 4.33526
Score 0.939605

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 21084 21084 0 0 0 - 2018-10-13 Solution found during MIPLIB2017 problem selection.

Similar instances in collection

The following instances are most similar to fastxgemm-n3r21s3t6 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-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
fastxgemm-n2r7s4t1 easy 904 56 0 848 6972 22584 Laurent Sorber fastxgemm 42 decomposition benchmark_suitable variable_bound set_partitioning set_covering mixed_binary general_linear
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
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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