On the complexity of binary polynomial optimization over acyclic hypergraphs
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In this work we consider binary polynomial optimization, which is the problem of maximizing a given polynomial function over all binary points. We completely settle the computational complexity of this problem over acyclic hypergraphs. Our main result is a strongly polynomial-time algorithm for β-acyclic hypergraphs. The idea behind our algorithm is to successively remove nest points from the hypergraph, until there is only one node left. Once we reach this condition, optimizing the problem becomes trivial. Then, we show that for α-acyclic hypergraphs the problem is strongly NP-hard, and NP-hard to approximate within a constant factor larger than 16/17 provided that P ≠ NP. Our algorithm can also be applied to any general binary polynomial optimization problem that contains β-cycles. For these problems, the algorithm returns a smaller instance together with a rule to extend any optimal solution of the smaller instance to an optimal solution of the original instance. We assess the effectiveness of this technique via computational experiments on random hypergraphs.
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