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. While polynomial-time approximation algorithms exist for restricted graph classes, general bounds remain loose or computationally expensive. In this paper, we exploit optimum cycle means (minimum and maximum cycle means), which are computable in strongly polynomial time, to derive both strict algebraic bounds and heuristic approximations for the weight and length of the longest simple cycle. We rigorously analyze the algebraic relationships between these mean statistics and the properties of longest cycles, and present dual results for shortest cycles. While the strict bounds provide polynomial-time computable constraints suitable for pruning search spaces in branch-and-bound algorithms, our proposed heuristic approximations offer precise estimates for the objective value. Experimental evaluation on ISCAS benchmark circuits demonstrates this trade-off: while the strict algebraic lower bounds are often loose (median 85--93% below true values), the heuristic approximations achieve median errors of only 6--14%. We also observe that maximum weight and maximum length cycles frequently coincide, suggesting that long cycles tend to accumulate large weights.

翻译：在有向图中寻找最长简单圈的问题是NP难的，在计算生物学、调度和网络分析中具有关键应用。虽然对于受限图类存在多项式时间近似算法，但一般界仍然宽松或计算代价高昂。本文利用可在强多项式时间内计算的最优圈均值（最小和最大圈均值），推导出最长简单圈权重和长度的严格代数界及启发式近似。我们严格分析了这些均值统计量与最长圈性质之间的代数关系，并给出了最短圈的对偶结果。严格界提供了适用于分支定界算法中搜索空间剪枝的多项式时间可计算约束，而所提出的启发式近似则为目标值提供了精确估计。在ISCAS基准电路上的实验评估展示了这种权衡：虽然严格代数下界通常较宽松（中位数低于真实值85-93%），但启发式近似的中位误差仅为6-14%。我们还观察到最大权重圈与最大长度圈经常重合，表明长圈倾向于积累较大权重。