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

Establish a hardness result for computing the matrix product X = AB in the supported low-bandwidth model when both input matrices A and B are general (potentially dense) and the set of required outputs is uniformly sparse (at most d nonzeros per row and column), i.e., the case GM × GM = US. Concretely, determine the communication or routing lower bounds needed to solve GM × GM = US, analogous to the hardness shown for US × GM = GM.

Background

The paper studies sparse matrix multiplication in the supported low-bandwidth model and provides conditional hardness results linking certain sparsity combinations to primitives such as broadcasting and routing. In the Routing subsection, the authors show that solving US × GM = GM requires substantial routing (at least one node must output Ω(√n) values held by other nodes), but they do not analyze the GM × GM = US case.

This unresolved case concerns the scenario where both inputs are fully general while the outputs of interest are uniformly sparse. Establishing a hardness result here would complete the analysis for the [US:GM:GM] family permutations and clarify whether similar routing bottlenecks occur when both inputs are dense.