A reduction from $HP$ to $\{(\langle M \rangle, \langle k \rangle) : \text{M visits in at list $k$ states for any input}\}$

I tried to define the next reduction from $ HP$ to $ \{(\langle M \rangle, \langle k \rangle) : \text{M visits in at list $ k$ states for any input}\}$ .

Given a couple $ (\langle M\rangle , \langle x\rangle)$ we define $ M_x$ such that for any input $ y$ , $ M_x$ simulates $ M$ on the input $ x$ . We denote $ Q_M +c$ the number of states needed for the simulation of $ M$ on $ x$ , and define more special states for $ M_x$ $ q_1′,q_2′,…,q_{Q_M + c+ 1}’$ when $ q’_{Q_M +c+1}$ is defined as the only final state of $ M_x$ . Now, in case $ M_x$ simulation of $ M$ on $ x$ halts (i.e $ M$ reach one of its finite state) $ M_x$ move to $ q_1’$ and then continue to walk through all the special states till it reaches $ q_{Q_M + c + 1}$ .

We define the reduction $ (\langle M \rangle , \langle x \rangle) \longrightarrow (\langle M_x \rangle , \langle Q_M +c+1 \rangle)$

In case $ ((\langle M \rangle , \langle x \rangle) \in HP$ then for any input $ y$ , $ M_x$ walks through all the special states and thus visits in at least $ Q_m + c+ 1$ steps. Otherwise, $ M$ doesn’t stop on $ x$ so $ M_x$ doesn’t visit any special state, thus visits at most $ Q_M +c$ states (the states needed for the simulation).

It is ok? If you have other ideas or suggestions please let me know.

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Find shortest path that visits all nodes in a given set of nodes

Suppose I have a graph $ G$ and a set of target nodes $ S = \{A_1, …, A_n\}$ . I’m attempting to find the shortest path that visits each target node, in order, without visiting the same node twice.

For example, consider the following:

example

I have attempted to solve this by first performing a breadth-first search to find the shortest path from A to B, then from B to C, and so on, taking care to exclude any paths already found at each iteration.

This greedy solution works for most inputs, but in some case it will fail to find a solution, as the shortest path between two target nodes may end up blocking the complete path.

Aside from a brute-force approach, is it possible to find the shortest path that visits all of the target nodes, and is guaranteed to find such a path if one exists? Any suggestions on how to proceed?

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