Submitter | Variables | Constraints | Density | Status | Group | Objective | MPS File |
---|---|---|---|---|---|---|---|
Laurent Sorber | 904 | 6972 | 3.58323e-03 | easy | fastxgemm | 41.99999999999998 | 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.
Detailed explanation of the following tables can be found here.
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 |
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 |
Available nonzero structure and decomposition information. Further information can be found here.
Decomposed structure of original problem (dec-file)
Decomposed structure after trivial presolving (dec-file)
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 |
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. |
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.
@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]"
}