Uniquely Solvable Puzzles and Fast Matrix Multiplication

• Main Author: Mebane, Palmer
• Format: Others
• Published: Scholarship @ Claremont 2012
• Subjects: 05B99 Designs and configurations - None of the above but in this section; 20C05 Group rings of finite groups and their modules; 68W30 Symbolic computation and algebraic computation
• Online Access: https://scholarship.claremont.edu/hmc_theses/37; https://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1036&context=hmc_theses

In 2003 Cohn and Umans introduced a new group-theoretic framework for doing fast matrix multiplications, with several conjectures that would imply the matrix multiplication exponent $\omega$ is 2. Their methods have been used to match one of the fastest known algorithms by Coppersmith and Winograd, which runs in $O(n^{2.376})$ time and implies that $\omega \leq 2.376$. This thesis discusses the framework that Cohn and Umans came up with and presents some new results in constructing combinatorial objects called uniquely solvable puzzles that were introduced in a 2005 follow-up paper, and which play a crucial role in one of the $\omega = 2$ conjectures.