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Fast Matrix Multiplication via Ternary Meta Flip Graphs
Perminov Andrew Igorevich
Matrix multiplication optimization remains a fundamental challenge in computational mathematics. This work introduces a novel approach that discovers matrix multiplication schemes whose coefficients are restricted to the set {−1,0,1} (denoted ZT), minimizing naive additive complexity for efficient hardware implementation. The core of the method is a GPU-accelerated meta flip graph algorithm that maintains ternary safety through specialized arithmetic operations and sign symmetry breaking. Key results include new best ranks for the formats 4×5×12, 5×6×10, and 6×7×9, the independent discovery of 32 schemes in ZT that match known optimal ranks (including 8 previously known only with rational coefficients), and 30 rank improvements in the binary field. The analysis of 164 known schemes shows that 92 admit a ternary-coefficient implementation, while 72 could not be found under this constraint, defining the current boundaries of the approach. All software, results, and discovered schemes are provided as open-source.