Submitter | Variables | Constraints | Density | Status | Group | Objective | MPS File |
---|---|---|---|---|---|---|---|

Laurent Sorber | 18684 | 219368 | 1.75214e-04 | open | fastxgemm | 4110.99800117485* | fastxgemm-n3r21s3t6.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 | 18684 | 18684 |

Constraints | 219368 | 219368 |

Binaries | 378 | 378 |

Integers | 0 | 1134 |

Continuous | 18306 | 17172 |

Implicit Integers | 0 | 1134 |

Fixed Variables | 0 | 0 |

Nonzero Density | 0.000175214 | 0.000175214 |

Nonzeroes | 718146 | 718146 |

Original | Presolved | |
---|---|---|

Total | 219368 | 219368 |

Empty | 0 | 0 |

Free | 0 | 0 |

Singleton | 0 | 0 |

Aggregations | 0 | 0 |

Precedence | 46494 | 0 |

Variable Bound | 46494 | 92988 |

Set Partitioning | 0 | 567 |

Set Packing | 0 | 0 |

Set Covering | 0 | 90 |

Cardinality | 0 | 0 |

Invariant Knapsack | 0 | 0 |

Equation Knapsack | 0 | 0 |

Bin Packing | 0 | 0 |

Knapsack | 0 | 0 |

Integer Knapsack | 0 | 0 |

Mixed Binary | 126380 | 2684 |

General Linear | 0 | 123039 |

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 | 1.342423 | ||||

Constraint % | 4.67707 | 4.67707 | 4.67707 | 4.67707 | |

Variable % | 4.33526 | 4.33526 | 4.33526 | 4.33526 | |

Score | 0.939605 |

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 | |
---|---|---|---|---|---|---|---|---|---|

3 | 4 | 4110.998 | 0 | 3e-07 | 0 | Edward Rothberg | 2019-12-13 | Obtained with Gurobi 9.0 | |

1 | 3 | 6084.000 | 6084 | 0 | 0e+00 | 0 | Frederic Didier | 2020-01-22 | Obtained with Google OR-tools using 8 Threads through generating subproblems by fixing part of the current solution and trying to solve them with a sub CP-SAT solver |

4 | 2 | 9084.000 | 0 | 0e+00 | 0 | Robert Ashford and Alkis Vazacopoulus | 2019-12-18 | Found using ODH|CPlex | |

2 | 1 | 21084.000 | 21084 | 0 | 0e+00 | 0 | - | 2018-10-13 | Solution found during MIPLIB2017 problem selection. |

The following instances are most similar to fastxgemm-n3r21s3t6 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]"
}
```

Last Update Nov 16, 2020 by Philipp Wellner

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