  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.
