An NP-complete graph decision problem, the "Multi-stage graph Simple Path" (abbr. MSP) problem, is introduced. The main contribution of this paper is a poly-time algorithm named the ZH algorithm for the problem together with the proof of its correctness, which implies NP=P. (1) A crucial structural property is discovered, whereby all MSP instances are arranged into the sequence $G_{0}$,$G_{1}$,$G_{2}$,... ($G_{k}$ essentially stands for a group of graphs for all $k\geq 0$). For each $G_{j}(j>0)$ in the sequence, there is a graph $G_{i}(0\leq i<j)$ "mathematically homomorphic" to $G_{j}$ which keeps completely accordant with $G_{j}$ on the existence of global solutions. This naturally provides a chance of applying mathematical induction for the proof of an algorithm. In previous attempts, algorithms used for making global decisions were mostly guided by heuristics and intuition. Rather, the ZH algorithm is dedicatedly designed to comply with the proposed proving framework of mathematical induction. (2) Although the ZH algorithm deals with paths, it always regards paths as a collection of edge sets. This is the key to the avoidance of exponential complexity. (3) Any poly-time algorithm that seeks global information can barely avoid the error caused by localized computation. In the ZH algorithm, the proposed reachable-path edge-set $R(e)$ and the computed information for it carry all necessary contextual information, which can be utilized to summarize the "history" and to detect the "future" for searching global solutions. (4) The relation between local strategies and global strategies is discovered and established, wherein preceding decisions can pose constraints to subsequent decisions (and vice versa). This interplay resembles the paradigm of dynamic programming, while much more convoluted. Nevertheless, the computation is always strait forward and decreases monotonically.

翻译：本文提出一个NP完全的图决策问题——"多阶段图简单路径"（简称MSP问题）。主要贡献是为该问题设计了一个名为ZH算法的多项式时间算法，并证明了其正确性，由此可得NP=P。(1) 发现一个关键结构性质：所有MSP实例可排列为序列$G_{0}$、$G_{1}$、$G_{2}$...（其中$G_{k}$本质上代表对所有$k\geq 0$的一组图）。对于该序列中每个$G_{j}(j>0)$，存在一个与$G_{j}$"数学同态"的图$G_{i}(0\leq i<j)$，且在全局解的存在性方面与$G_{j}$保持完全一致。这为算法证明提供了自然运用数学归纳法的机会。以往用于全局决策的算法多受启发式和直觉引导，而ZH算法则专门设计以契合所提出的数学归纳证明框架。(2) 尽管ZH算法处理路径，但始终将路径视为边集的集合，这是规避指数复杂度的关键。(3) 任何寻求全局信息的多项式时间算法都难以避免局部化计算导致的误差。在ZH算法中，所提出的可达路径边集$R(e)$及其计算信息携带所有必要的上下文信息，可用来总结"历史"并探测"未来"以搜索全局解。(4) 发现了局部策略与全局策略的关系并建立联系：前置决策约束后续决策（反之亦然）。这种相互作用类似动态规划范式但更为复杂，然而计算始终是直接的且单调递减。