Node-Weighted Triangles: Faster and Simpler

2026-05-08

AI Summary

This work addresses the problem of detecting a triangle whose vertex weights sum to zero in a vertex-weighted graph. The authors propose an algebraic algorithm based on matrix multiplication, denoted MM(n), that solves the problem in O(MM(n)) time—matching the best-known time complexity for unweighted triangle detection and achieving the first such bound for the weighted variant. This result not only attains the theoretically optimal running time but also eliminates the super-logarithmic factors present in prior approaches. Furthermore, the algorithm substantially simplifies the overall design by avoiding intricate recursive structures and communication protocols, thereby offering both computational efficiency and implementation clarity.

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 $2^{Ω(\sqrt{\log n})}$ 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.

Problem

• Node-Weighted Triangle
• triangle detection
• fine-grained complexity
• vertex-weighted graph
• zero-sum triangle

Innovation

• Node-Weighted Triangle
• Matrix Multiplication Time
• Fine-Grained Complexity
• Triangle Detection
• Algorithmic Simplicity