Submitter Variables Constraints Density Status Group Objective MPS File
Laurent Sorber 20394 240116 1.60591e-04 open fastxgemm 27087 fastxgemm-n3r23s5t6.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 20394 20394
Constraints 240116 240116
Binaries 414 414
Integers 0 1242
Continuous 19980 18738
Implicit Integers 0 1242
Fixed Variables 0 0
Nonzero Density 0.000160591 0.000160591
Nonzeroes 786402 786402
Constraint Classification Properties
Original Presolved
Total 240116 240116
Empty 0 0
Free 0 0
Singleton 0 0
Aggregations 0 0
Precedence 50922 0
Variable Bound 50922 101844
Set Partitioning 0 621
Set Packing 0 0
Set Covering 0 96
Cardinality 0 0
Invariant Knapsack 0 0
Equation Knapsack 0 0
Bin Packing 0 0
Knapsack 0 0
Integer Knapsack 0 0
Mixed Binary 138272 2798
General Linear 0 134757
Indicator 0 0

Structure

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

value min median mean max
Components 1.380211
Constraint % 4.27293 4.27293 4.27293 4.27293
Variable % 3.97176 3.97176 3.97176 3.97176
Score 0.943742

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 27087 27087 0 0 0 - 2018-10-13 Solution found during MIPLIB2017 problem selection.

Similar instances in collection

The following instances are most similar to fastxgemm-n3r23s5t6 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-n3r22s4t6 19539 396 0 19143 229742 752274 Laurent Sorber fastxgemm open 21084*
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 17, 2018 by Gregor Hendel
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