Bounds on Longest Simple Cycles in Weighted Directed Graphs via Optimum Cycle Means

2025-12-31

arXiv.org

AI Summary

Finding the longest simple cycle in a directed graph is NP-hard, and existing bounds are either too loose or computationally expensive. This work proposes the first tight algebraic bounds on the weight and length of the longest simple cycle by leveraging the optimal cycle mean (maximum or minimum), which can be computed in strongly polynomial time. Building upon these bounds, we develop a high-accuracy heuristic approximation. Through algebraic analysis, we reveal that maximum-weight cycles often substantially overlap with longest cycles, a property we exploit within a branch-and-bound framework to enhance pruning efficiency. Experimental evaluation on ISCAS benchmark circuits demonstrates that the proposed provable lower bound achieves median errors of 85–93%, while the heuristic approximation reduces this error dramatically to only 6–14%, significantly outperforming existing approaches.

Technology Category

• Search and Optimization: Heuristic Search
• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

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

The problem of finding the longest simple cycle in a directed graph is NP-hard, with critical applications in computational biology, scheduling, and network analysis. Existing approaches include exact algorithms with exponential runtimes, approximation algorithms limited to specific graph classes, and heuristics with no formal guarantees. In this paper, we exploit optimum cycle means (minimum and maximum cycle means), computable in strongly polynomial time, to derive both strict bounds and heuristic estimates for the weight and length of the longest simple cycle in general graphs. The strict bounds can prune search spaces in exact algorithms while the heuristic estimates (the arithmetic mean and geometric mean of the optimum cycle means) guarantee bounded approximation error. Crucially, a single computation of optimum cycle means yields both the bounds and the heuristic estimates. Experimental evaluation on ISCAS benchmark circuits demonstrates that, compared to true values, the strict algebraic lower bounds are loose (median 80--99% below) while the heuristic estimates are much tighter: the arithmetic mean and the geometric mean have median errors of 6--13% vs. 11--21% for symmetric (uniform) weights and 41--92% vs. 25--35% for skewed (log-normal) weights, favoring the arithmetic mean for symmetric distributions and the geometric mean for skewed distributions.

Problem

• longest simple cycle
• weighted directed graphs
• optimum cycle means
• NP-hard
• cycle bounds

Innovation

• optimum cycle means
• longest simple cycle
• weighted directed graphs
• algebraic bounds
• heuristic approximation