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.

Detailed explanation of the following tables can be found here.

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 |

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 |

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 | 1.380211 | ||||

Constraint % | 4.27293 | 4.27293 | 4.27293 | 4.27293 | |

Variable % | 3.97176 | 3.97176 | 3.97176 | 3.97176 | |

Score | 0.943742 |

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. |

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* |

```
@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 Jan 18, 2019 by Gregor Hendel

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