The Limits of Black-Box Reductions for All-Pairs Triangle Detection

For any tripartite relation $R\subseteq \mathbb{Z}^3$, the $R$-Triangle problem asks, given an edge-weighted graph, whether it contains a triangle whose weights form a triple in $R$. The All-Edge $R$-Triangle problem asks to determine for every edge whether it is contained in such a triangle. It is known that $R$-Triangle and All-Edge $R$-Triangle are subcubically fine-grained equivalent for every $R$ [Vassilevska W.-Williams'10]. However, while it is conjectured that these problems are tightly equivalent, this reduction only shows that if $R$-Triangle has an $O(n^{3-ε})$-time algorithm for some $ε>0$, then All-Edge $R$-Triangle has an $O(n^{3-ε/3})$-time algorithm. This paper provides a strong unconditional barrier to a tight equivalence: the reduction of [Vassilevska W.-Williams'10] is optimal for black-box reductions that work for arbitrary $R$. We give further results about black-box reductions between a variety of $R$-triangle problems. Our positive results yield new reductions between several classes of triangle and matrix problems --- for instance, we demonstrate that an $O(n^{2.53})$-time algorithm for computing equality or dominance product would imply an improvement on known algorithms for computing boolean $(\min, +)$-product, giving the first conditional lower bound for dominance and equality product. Our negative results can be thought of as barriers against natural fine-grained proof techniques. Besides the result that a tighter equivalence between $R$-Triangle and All-Edge $R$-Triangle is not possible, we also show that no appropriately "black-box" reductions are capable of demonstrating a subcubic equivalence between triangle counting and binary integer matrix multiplication, or a tight equivalence between boolean matrix multiplication and listing $n^2$ triangles, and more, despite the fact that all of these equivalences are conjectured to hold.