A New General-Purpose Method to Multiply 3x3 Matrices Using Only 23 Multiplications

One of the most famous conjectures in computer algebra is that matrix multiplication might be feasible in not much more than quadratic time. The best known exponent is 2.376, due to Coppersmith and Winograd. Many attempts to solve this problems in the literature work by solving, fixed-size problems and then apply the solution recursively. This leads to pure combinatorial optimisation problems with fixed size. These problems are unlikely to be solvable in polynomial time. In 1976 Laderman published a method to multiply two 3x3 matrices using only 23 multiplications. This result is non-commutative, and therefore can be applied recursively to smaller sub-matrices. In 35 years nobody was able to do better and it remains an open problem if this can be done with 22 multiplications. We proceed by solving the so called Brent equations [7]. We have implemented a method to converting this very hard problem to a SAT problem, and we have attempted to solve it, with our portfolio of some 500 SAT solvers. With this new method we were able to produce new solutions to the Laderman's problem. We present a new fully general non-commutative solution with 23 multiplications and show that this solution is new and is NOT an equivalent variant of the Laderman's original solution. This result demonstrates that the space of solutions to Laderman's problem is larger than expected, and therefore it becomes now more plausible that a solution with 22 multiplications exists. If it exists, we might be able to find it soon just by running our algorithms longer, or due to further improvements in the SAT solver algorithms.Comment: This work was supported by the UK Technology Strategy Board under Project No: 9626-5852