On the Complexity of Binary Polynomial Optimization Over Acyclic Hypergraphs

In this work, we advance the understanding of the fundamental limits of computation for binary polynomial optimization (BPO), which is the problem of maximizing a given polynomial function over all binary points. In our main result we provide a novelclass of BPO that can be solved efficiently both from a theoretical and computational perspective. In fact, we give a strongly polynomial-time algorithm for instances whosec orresponding hypergraph is beta-acyclic. We note that the beta-acyclicity assumption isnatural in several applications including relational database schemes and the liftedmulticut problem on trees. Due to the novelty of our proving technique, we obtainan algorithm which is interesting also from a practical viewpoint. This is becauseour algorithm is very simple to implement and the running time is a polynomial ofvery low degree in the number of nodes and edges of the hypergraph. Our result completely settles the computational complexity of BPO over acyclic hypergraphs,since the problem is NP-hard on alpha-acyclic instances. Our algorithm can also be appliedto any general BPO problem that contains beta-cycles. For these problems, the algorithmreturns a smaller instance together with a rule to extend any optimal solution of thesmaller instance to an optimal solution of the original instance.