A Fast Algorithm for Finding Minimum Weight Cycles in Mining Cyclic Graph Topologies

2025-11-30

AI Summary

Detecting the minimum-weight cycle (MWC) in weighted graphs suffers from high computational complexity and a lack of efficient deterministic algorithms. To address this, we propose a novel node-iterative optimization algorithm grounded in the Dijkstra framework. Our key contributions are: (1) a composite distance metric integrating path weight and cycle-closure cost; (2) a provably correct node pruning criterion leveraging loop invariants for dynamic elimination of suboptimal intermediate results; and (3) a local-structure-aware subgraph pruning heuristic. The algorithm guarantees global optimality while significantly reducing time complexity. Extensive experiments on diverse real-world graphs demonstrate speedups of 1.5×–5.8× over state-of-the-art methods. Moreover, it enables efficient computation of higher-order network features such as Loop Modulus, underscoring its practical utility in large-scale graph analytics.

Technology Category

• Search and Optimization: Algorithm Configuration
• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Cyclic structures are fundamental topological features in graphs, playing critical roles in network robustness, information flow, community structure, and various dynamic processes. Algorithmic tools that can efficiently probe and analyze these cyclic topologies are increasingly vital for tasks in graph mining, network optimization, bioinformatics, and social network analysis. A core primitive for quantitative analysis of cycles is finding the Minimum Weight Cycle (MWC), representing the shortest cyclic path in a weighted graph. However, computing the MWC efficiently remains a challenge, particularly compared to shortest path computations. This paper introduces a novel deterministic algorithm for finding the MWC in general weighted graphs. Our approach adapts the structure of Dijkstra's algorithm by introducing and minimizing a extit{composite distance} metric, effectively translating the global cycle search into an iterative node-centric optimization. We provide a rigorous proof of correctness based on loop invariants. We detail two mechanisms for accelerating the search: a provable node discarding technique based on intermediate results, and a highly effective graph pruning heuristic. This heuristic dynamically restricts the search to relevant subgraphs, leveraging the principle of locality often present in complex networks to achieve significant empirical speedups, while periodic resets ensure global optimality is maintained. The efficiency of the proposed MWC algorithm enables its use as a core component in more complex analyses focused on cyclic properties. We illustrate this through a detailed application case study: accelerating the computation of the Loop Modulus, a measure of cycle richness used in advanced network characterization. Our algorithm dramatically reduces the runtime of the iterative constraint-finding bottleneck in this computation.

Problem

• Develops a fast deterministic algorithm for finding Minimum Weight Cycles in weighted graphs
• Accelerates search via node discarding and dynamic graph pruning heuristics
• Enables efficient analysis of cyclic properties in complex networks

Innovation

• Adapts Dijkstra's algorithm with composite distance metric
• Uses node discarding and graph pruning for acceleration
• Enables efficient computation of Minimum Weight Cycle in graphs