Submitter Variables Constraints Density Status Group Objective MPS File
Laurent Sorber 19539 229742 1.67584e-04 open fastxgemm 21084 fastxgemm-n3r22s4t6.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 19539 19539
Constraints 229742 229742
Binaries 396 396
Integers 0 1188
Continuous 19143 17955
Implicit Integers 0 1188
Fixed Variables 0 0
Nonzero Density 0.000167584 0.000167584
Nonzeroes 752274 752274
Constraint Classification Properties
Original Presolved
Total 229742 229742
Empty 0 0
Free 0 0
Singleton 0 0
Aggregations 0 0
Precedence 48708 0
Variable Bound 48708 97416
Set Partitioning 0 594
Set Packing 0 0
Set Covering 0 93
Cardinality 0 0
Invariant Knapsack 0 0
Equation Knapsack 0 0
Bin Packing 0 0
Knapsack 0 0
Integer Knapsack 0 0
Mixed Binary 132326 2741
General Linear 0 128898
Indicator 0 0

Structure

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

value min median mean max
Components 1.361728
Constraint % 4.46588 4.46588 4.46588 4.46588
Variable % 4.14556 4.14556 4.14556 4.14556
Score 0.941764

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-n3r22s4t6 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 Variables Binaries Integers Continuous Constraints Nonz. Submitter Group Status Objective
fastxgemm-n3r23s5t6 20394 414 0 19980 240116 786402 Laurent Sorber fastxgemm open 27087*
fastxgemm-n3r21s3t6 18684 378 0 18306 219368 718146 Laurent Sorber fastxgemm open 21084*
fastxgemm-n2r7s4t1 904 56 0 848 6972 22584 Laurent Sorber fastxgemm easy 42
fastxgemm-n2r6s0t2 784 48 0 736 5998 19376 Laurent Sorber fastxgemm easy 230
neos-4335793-snake 30827 20473 7865 2489 37166 129119 Jeff Linderoth neos-pseudoapplication-44 open 43.00000000009*

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]"
}

Last Update Nov 09, 2018 by Gregor Hendel
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