Non-constant cost of Integer Inner Product (IIP_d)

Determine whether, for every fixed dimension d, the Integer Inner Product problem IIP_d requires non-constant public-coin randomized communication cost; equivalently, show that IIP_d does not admit constant-cost randomized protocols and thus does not belong to the constant-cost class BPP defined in this paper.

Background

The Integer Inner Product problems IIP_d are known to have randomized communication complexity O(d * log n) and thus lie in the standard communication class BPP (as the union over polylogarithmic costs). However, it is widely believed they do not admit constant-cost protocols. The authors use IIP_d to illustrate separations within constant-cost reductions and emphasize that resolving the conjectured non-constancy would further clarify the landscape.