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Hardness for BD × GM = BD and GM × BD = BD in the supported low-bandwidth model

Establish hardness results for computing the matrix product X = AB in the supported low-bandwidth model for the cases BD × GM = BD and GM × BD = BD, where BD denotes matrices of bounded degeneracy and GM denotes general matrices. Precisely, determine the communication or routing lower bounds required when one input is BD and the other is GM, and the outputs of interest are BD.

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Background

The paper proves a hardness result for BD × BD = GM (via the special case RS × CS = GM) showing that at least one node must output Ω(√n) values originally held by other nodes. However, the authors explicitly leave the mixed BD/GM cases—BD × GM = BD and GM × BD = BD—unresolved.

Resolving these cases would extend the hardness landscape for combinations involving bounded degeneracy, clarifying whether similar lower bounds apply when only one of the inputs is BD and the other is fully general.

References

We prove a hardness result for $BD \times BD = GM$; the case of $BD \times GM = BD$ and $GM \times BD = BD$ is left for future work.

— Low-Bandwidth Matrix Multiplication: Faster Algorithms and More General Forms of Sparsity (2404.15559 - Gupta et al., 23 Apr 2024) in Section 6.3 (Routing)