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Recently, several classes of cutting planes have been introduced for binary polynomial optimization. In this paper, we present the first results connecting the combinatorial structure of these inequalities with their Chvátal rank. We determine the Chvátal rank of all known cutting planes and show that almost all of them have Chvátal rank 1. We observe that these inequalities have an associated hypergraph that is β-acyclic. Our second goal is to derive deeper cutting planes; to do so, we consider hypergraphs that admit β-cycles. We introduce a novel class of valid inequalities arising from odd β-cycles, that generally have Chvátal rank 2. These inequalities allow us to obtain the first characterization of the multilinear polytope for hypergraphs that contain β-cycles. Namely, we show that the multilinear polytope for cycle hypergraphs is given by the standard linearization inequalities, flower inequalities, and odd β-cycle inequalities. We also prove that odd β-cycle inequalities can be separated in linear time when the hypergraph is a cycle hypergraph. This shows that instances represented by cycle hypergraphs can be solved in polynomial time. Last, to test the strength of odd β-cycle inequalities, we perform numerical experiments that imply that they close a significant percentage of the integrality gap.
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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 novel class 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 whose corresponding hypergraph is β-acyclic. We note that the β-acyclicity assumption is natural in several applications including relational database schemes and the lifted multicut problem on trees. Due to the novelty of our proving technique, we obtain an algorithm which is interesting also from a practical viewpoint. This is because our algorithm is very simple to implement and the running time is a polynomial of very 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 α-acyclic instances. Our algorithm can also be applied to any general BPO 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.
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- Award ID(s):
- 1634768
- PAR ID:
- 10330008
- Date Published:
- Journal Name:
- Proceedings of the Annual ACMSIAM Symposium on Discrete Algorithms
- ISSN:
- 1557-9468
- Page Range / eLocation ID:
- 2684 - 2699
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1137/1.9781611977073.105
