Ant Colony Optimization With Combining Gaussian Eliminations for Matrix Multiplication

One of the main unsolved problems in computer algebra is to determine the minimal number of multiplications which is necessary to compute the product of two matrices. For practical value, the small format is of special interest. This leads to a combinatorial optimization problem which is unlikely solved in polynomial time. In this paper, we present a method called combining Gaussian eliminations to reduce the number of variables in this optimization problem and use heuristic ant colony algorithm to solve the problem. The results of experiments on 2 × 2 case show that our algorithm achieves significant performance gains. Extending this algorithm from 2 × 2 case to 3 × 3 case is also discussed.