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Given a general branching program, is there an algorithm which can find an equivalent branching program $P$ of minimal length. That is $|P| \leq |P'|$ for all equivalent branching program $P'$.

If not, are their some specific cases where this algorithm exists?

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I'm assuming you mean branching programs in the sense of generalized binary decision diagrams (see here).

Now the question is whether by "length", you mean "size" or "depth". I will answer the "size" version here, as you will most likely mean that (and the depth version has some intricacies).

We can represent a smaller solution in space not larger than the input size. Furthermore, checking two branching programs to be equivalent is a problem contained in the complexity class co-NP.

Hence, we can define an algorithm in $\Sigma^P_2$ (see here) for finding if there exists a smaller equivalent branching program by letting it first guess a smaller solution, and then verify it to be correct using the co-NP equivalence checking oracle.

Note that this may not be an algorithm that is complexity-theoretically optimal. However, the existence of this algorithm answers your question with a "yes".

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