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.

翻译：在有向图中寻找最长简单环的问题是NP难的，在计算生物学、调度和网络分析中具有关键应用。现有方法包括具有指数运行时间的精确算法、仅限于特定图类的近似算法以及无形式化保证的启发式方法。本文利用可在强多项式时间内计算的最优环均值（最小和最大环均值），推导出一般图中最长简单环权重和长度的严格界与启发式估计。严格界可用于精确算法中的搜索空间剪枝，而启发式估计（最优环均值的算术平均与几何平均）则保证有界的近似误差。关键在于，单次最优环均值计算即可同时获得界与启发式估计。在ISCAS基准电路上的实验评估表明，与真实值相比，严格代数下界较为宽松（中位数低于真实值80–99%），而启发式估计则更为精确：对于对称（均匀）权重，算术平均与几何平均的中位数误差分别为6–13%和11–21%；对于偏斜（对数正态）权重，分别为41–92%和25–35%，其中对称分布更适用算术平均，偏斜分布更适用几何平均。