Submitter Variables Constraints Density Status Group Objective MPS File
Laurent Sorber 784 5998 4.12042e-03 easy fastxgemm 230 fastxgemm-n2r6s0t2.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 784 784
Constraints 5998 5998
Binaries 48 48
Integers 0 144
Continuous 736 592
Implicit Integers 0 144
Fixed Variables 0 0
Nonzero Density 0.00412042 0.00412042
Nonzeroes 19376 19376
Constraint Classification Properties
Original Presolved
Total 5998 5998
Empty 0 0
Free 0 0
Singleton 0 0
Aggregations 0 0
Precedence 1224 0
Variable Bound 1224 2448
Set Partitioning 0 72
Set Packing 0 0
Set Covering 0 30
Cardinality 0 0
Invariant Knapsack 0 0
Equation Knapsack 0 0
Bin Packing 0 0
Knapsack 0 0
Integer Knapsack 0 0
Mixed Binary 3550 304
General Linear 0 3144
Indicator 0 0

Structure

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

value min median mean max
Components 0.845098
Constraint % 15.4385 15.4385 15.4385 15.4385
Variable % 12.7551 12.7551 12.7551 12.7551
Score 0.808157

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
1 230 230 0 0 0 - 2018-10-13 Solution found during MIPLIB2017 problem selection.

Similar instances in collection

The following instances are most similar to fastxgemm-n2r6s0t2 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-n2r7s4t1 904 56 0 848 6972 22584 Laurent Sorber fastxgemm easy 42
fastxgemm-n3r21s3t6 18684 378 0 18306 219368 718146 Laurent Sorber fastxgemm open 21084*
fastxgemm-n3r22s4t6 19539 396 0 19143 229742 752274 Laurent Sorber fastxgemm open 21084*
fastxgemm-n3r23s5t6 20394 414 0 19980 240116 786402 Laurent Sorber fastxgemm open 27087*
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 19, 2018 by Gregor Hendel
generated with R Markdown
© 2018 by Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB)
Imprint