Combinatorial Matrix Multiplication Conjecture

Determine whether, for all positive \alpha, \beta, and \gamma, no combinatorial algorithm can multiply an n^\alpha\times n^\beta matrix by an n^\beta\times n^\gamma matrix in time O(n^{\alpha+\beta+\gamma-\varepsilon}) for any \varepsilon>0.

Background

The paper introduces the Combinatorial Matrix Multiplication Conjecture as a conditional hardness assumption for the Two Intervals Matching Problem. In that problem, a data structure preprocesses two point sets and must answer queries involving two potentially distinct intervals containing equal numbers of red and blue points.

The conjecture is used to rule out combinatorial data structures with simultaneously low preprocessing and query costs for this two-interval matching problem. The reduction proceeds through the Hamming Distance Oracle Problem, whose hardness follows from the conjectured limitations of combinatorial matrix multiplication.

References

Our lower bound is based on the following conjecture that has been stated many times (#1{see, e.g.,)}: For any \alpha, \beta, \gamma, > 0, there is no combinatorial algorithm for multiplying an n\alpha \times n\beta matrix with an n\beta \times n\gamma matrix in time O(n{\alpha+\beta+\gamma-}).

— Computing All Optimal Partial $p$-Wasserstein Matchings on the Line  (2608.18875 - Angrick et al., 19 Aug 2026) in Appendix, Section Omitted Lower Bounds, subsection Two Intervals Matching Problem