Node-Weighted Triangles: Faster and Simpler

Weighted variants of triangle detection are an important object of study because of their prominence in fine-grained complexity. We revisit the Node-Weighted Triangle problem, where the goal is to decide if a vertex-weighted graph contains a triangle whose node weights sum to zero. This problem has been the focus of a celebrated line of work, beginning with a subcubic-time algorithm [Vassilevska, Williams; STOC '06], and culminating in algorithms running almost in matrix multiplication time, $O(\mathsf{\text{MM}}(n)+n^2\cdot 2^{O(\sqrt{\log n})})$ [Czumaj, Lingas; SODA '07], [Vassilevska W., Williams; STOC '09]. This runtime is almost-optimal, since even detecting an unweighted triangle is conjectured to require matrix multiplication time $\mathsf{\text{MM}}(n)$. However, the superpolylogarithmic $2^{\mathrm{\Omega }(\sqrt{\log n})}$ overhead persists in a world where near-optimal matrix multiplication is possible (i.e., $\mathsf{\text{MM}}(n)\le n^2\text{poly}(\log n)$). In this paper, we present a new algorithm solving Node-Weighted Triangle in $O(\mathsf{\text{MM}}(n))$ time, closing the gap to unweighted triangle detection completely. Remarkably, our algorithm is much simpler than previous approaches, which use involved recursion schemes and communication protocols.