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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The Running Intersection Relaxation of the Multilinear Polytope
The multilinear polytope of a hypergraph is the convex hull of a set of binary points satisfying a collection of multilinear equations. We introduce the running intersection inequalities, a new class of facet-defining inequalities for the multilinear polytope. Accordingly, we define a new polyhedral relaxation of the multilinear polytope, referred to as the running intersection relaxation, and identify conditions under which this relaxation is tight. Namely, we show that for kite-free beta-acyclic hypergraphs, a class that lies between gamma-acyclic and beta-acyclic hypergraphs, the running intersection relaxation coincides with the multilinear polytope and it admits a polynomial size extended formulation.
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- Award ID(s):
- 1634768
- NSF-PAR ID:
- 10210846
- Date Published:
- Journal Name:
- Mathematics of Operations Research
- Volume:
- 46
- Issue:
- 3
- ISSN:
- 0364-765X
- Page Range / eLocation ID:
- 1008 to 1037
- 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
- Journal Article:
- https://doi.org/10.1287/moor.2021.1121
