AI 中文总结
研究有向图中长度受限的简单路径和环的枚举问题，指出CYCLE_SEARCH和BC-DFS算法缺陷，引入边一致概念，设计分析新的BS-DFS算法及其变体，并指出BC-DFS失败机制，实验验证其遗漏非罕见。
AI 中文摘要
枚举受给定长度限制的简单路径和环是图算法中的一个基本问题，应用广泛。近期算法CYCLE_SEARCH和BC-DFS使用屏障值来修剪无果搜索，但都不完整且延迟边界论证有缺陷。本文通过新反例和识别BC-DFS屏障更新过程中的缺陷得出类似结果。主要贡献是引入边一致概念，以此为统一框架设计分析有界范围深度优先搜索（BS-DFS）及其变体，还用其指出BC-DFS的确切失败机制。实验结果证实BC-DFS的遗漏并非孤立边缘情况，在随机图上出现频率显著。
英文摘要
Enumerating simple paths and cycles subject to a given length bound is a fundamental problem in graph algorithms. Recent algorithms, namely BC-DFS (Peng et al. 2019, 2021) and CYCLE_SEARCH (Gupta and Suzumura 2021, arXiv:2105.10094v2), employ cached barrier values to prune fruitless searches. Both algorithms turn out to produce incomplete output, and their delay-bound arguments rely on flawed claims. For CYCLE_SEARCH this is known (arXiv:2512.08392); here we establish the analogous results for BC-DFS by exhibiting graphs on which paths are missed, by identifying the defect in its barrier-update procedure, and by refuting the monotonicity claim on which its delay-bound proof rests. As our main contribution, we introduce edge-consistency, a local invariant on barrier values analogous to heuristic consistency in informed search. It provides an incremental mechanism for maintaining admissible barrier estimates and yields concise correctness proofs. We use edge-consistency as a unifying framework for design and analysis of Bounded-Scope Depth-First Search (BS-DFS) --- a new algorithm for enumerating simple paths or cycles of length at most $k$ in a directed graph. For BS-DFS we prove a worst-case delay of at most $3(k+1)(n+m)$ elementary steps between consecutive events (start, each output, termination) and an amortized delay of at most $2(k+1)(n+m)$ steps per event, the $p$-th event being reached within $2p(k+1)(n+m)$ steps; both bounds are in $O(k(n+m))$. Barrier admissibility alone is not sufficient for the delay bound: for two variants with simpler barrier management, we exhibit a graph family forcing $Ω(k^2(n+m))$ delay between outputs. Experiments on two families of random graphs confirm our findings, support the significance of the incompleteness result, and show that achieving completeness has modest empirical cost.
CommentsSubmitted to the Journal of Graph Algorithms and Applications (JGAA)
