Node-Weighted Triangles: Faster and Simpler

TL;DR

This paper introduces a simpler, faster algorithm for the Node-Weighted Triangle problem, achieving near-optimal runtime by matching matrix multiplication time and closing a longstanding complexity gap.

Contribution

The authors present a novel, simpler algorithm that solves Node-Weighted Triangle in $O( extsf{MM}(n))$ time, improving over previous complex methods.

Findings

New algorithm runs in $O( extsf{MM}(n))$ time.

Closes the complexity gap to unweighted triangle detection.

Simplifies previous approaches using recursion and communication protocols.

Abstract

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, $O(\textsf{MM}(n) + n^2\cdot 2^{O(\sqrt{\log n})})$ [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 $\textsf{MM}(n)$. However, the superpolylogarithmic $2^{\Omega(\sqrt{\log n})}$ overhead persists in a world where near-optimal matrix multiplication is possible (i.e., $\textsf{MM}(n) \leq n^2\text{poly}(\log n)$). In this paper, we present a new algorithm solving Node-Weighted Triangle in $O(\textsf{MM}(n))$ 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.