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EDIT: Added an answer meeting the unique-sum requirement. Lemma 1. The problem is NP-hard by reduction from 3-CNF-SAT, even if the maximum is required to be unique. Proof. Here's the reduction. First we describe the reduction to the problem without the requirement that the maximum is unique. Fix a 3-CNF-SAT instance $\phi$. Assume WLOG that $\phi$ has more ...