HAM is NP-complete. First, we'll show that deciding if a directed graph has a Hamiltonian circuit is NP hard. We'll reduce 3-CNF-SAT to it.
Idea: We have a gadget for each clause and an assignment-selection gadget that makes us choose Boolean values for each of the variables one at a time.
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Now, we can turn any undirected instance into a directed instance by expanding each node into a trio of nodes: for incoming edges, outgoing edges, and a middle node to force us to go from in to out. Why do we need the middle node?
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