Combinatorial equivalence between triangle counting and all-edge triangle counting

Establish a combinatorial fine-grained subcubic equivalence between triangle counting and all-edge triangle counting, thereby determining whether fast algorithms for triangle counting generically imply comparably fast algorithms for all-edge triangle counting and integer matrix multiplication.

Background

The paper relates all-edge triangle counting to binary integer matrix multiplication and explains that integer matrix multiplication over polynomially bounded integers can be reduced to a polylogarithmic number of binary matrix multiplications. Binary matrix multiplication is equivalent to all-edge triangle counting, so a reduction from all-edge triangle counting to ordinary triangle counting would yield a corresponding reduction from integer matrix multiplication to triangle counting.

The authors distinguish the desired result from a trivial subcubic reduction obtained by directly solving both problems. They specifically seek a combinatorial reduction analogous to the known detection-versus-all-edge-detection equivalence. The paper proves a black-box barrier ruling out a broad class of such reductions, but does not resolve whether a non-black-box combinatorial equivalence exists.