The Boolean polynomial polytope with multiple choice constraints
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| Format: | Preprint |
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2024
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| author | Shao, Sihong Wu, Yishan |
| author_facet | Shao, Sihong Wu, Yishan |
| contents | We consider a class of $0$-$1$ polynomial programming termed multiple choice polynomial programming (MCPP) where the constraint requires exact one component per subset of the partition to be $1$ after all the entries are partitioned. Compared to the unconstrained counterpart, there are few polyhedral studies of MCPP in general form. This paper serves as the first attempt to propose a polytope associated with a hypergraph to study MCPP, which is the convex hull of $0$-$1$ vectors satisfying multiple choice constraints and production constraints. With the help of the decomposability property, we obtain an explicit half-space representation of the MCPP polytope when the underlying hypergraph is $α$-acyclic by induction on the number of hyperedges, which is an analogy of the acyclicity results on the multilinear polytope by Del Pia and Khajavirad (SIAM J Optim 28 (2018) 1049) when the hypergraph is $γ$-acyclic. We also present a necessary and sufficient condition for the inequalities lifted from the facet-inducing ones for the multilinear polytope to be still facet-inducing for the MCPP polytope. This result covers the particular cases by Bärmann, Martin and Schneider (SIAM J Optim 33 (2023) 2909). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_14207 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Boolean polynomial polytope with multiple choice constraints Shao, Sihong Wu, Yishan Optimization and Control Combinatorics 90C09, 52B12, 90C57, 05C65, 90C26 We consider a class of $0$-$1$ polynomial programming termed multiple choice polynomial programming (MCPP) where the constraint requires exact one component per subset of the partition to be $1$ after all the entries are partitioned. Compared to the unconstrained counterpart, there are few polyhedral studies of MCPP in general form. This paper serves as the first attempt to propose a polytope associated with a hypergraph to study MCPP, which is the convex hull of $0$-$1$ vectors satisfying multiple choice constraints and production constraints. With the help of the decomposability property, we obtain an explicit half-space representation of the MCPP polytope when the underlying hypergraph is $α$-acyclic by induction on the number of hyperedges, which is an analogy of the acyclicity results on the multilinear polytope by Del Pia and Khajavirad (SIAM J Optim 28 (2018) 1049) when the hypergraph is $γ$-acyclic. We also present a necessary and sufficient condition for the inequalities lifted from the facet-inducing ones for the multilinear polytope to be still facet-inducing for the MCPP polytope. This result covers the particular cases by Bärmann, Martin and Schneider (SIAM J Optim 33 (2023) 2909). |
| title | The Boolean polynomial polytope with multiple choice constraints |
| topic | Optimization and Control Combinatorics 90C09, 52B12, 90C57, 05C65, 90C26 |
| url | https://arxiv.org/abs/2405.14207 |