Strengthening convex relaxations of 0/1-sets using Boolean formulas

In convex integer programming, various procedures have been developed to strengthen convex relaxations of sets of integer points. On the one hand, there exist several general-purpose methods that strengthen relaxations without specific knowledge of the set S of feasible integer points, such as popul...

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Vydané v:	Mathematical programming Ročník 190; číslo 1-2; s. 467 - 482
Hlavní autori:	Fiorini, Samuel, Huynh, Tony, Weltge, Stefan
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2021 Springer Nature B.V
Predmet:	Boolean Boolean algebra Calculus of Variations and Optimal Control; Optimization Combinatorial analysis Combinatorics Full Length Paper Hierarchies Integer programming Linear programming Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Optimization Theoretical 90Cxx 68Q06 52Bxx
ISSN:	0025-5610, 1436-4646
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Abstract	In convex integer programming, various procedures have been developed to strengthen convex relaxations of sets of integer points. On the one hand, there exist several general-purpose methods that strengthen relaxations without specific knowledge of the set S of feasible integer points, such as popular linear programming or semi-definite programming hierarchies. On the other hand, various methods have been designed for obtaining strengthened relaxations for very specific sets S that arise in combinatorial optimization. We propose a new efficient method that interpolates between these two approaches. Our procedure strengthens any convex set containing a set S ⊆ { 0 , 1 } n by exploiting certain additional information about S . Namely, the required extra information will be in the form of a Boolean formula ϕ defining the target set S . The new relaxation is obtained by “feeding” the convex set into the formula ϕ . We analyze various aspects regarding the strength of our procedure. As one application, interpreting an iterated application of our procedure as a hierarchy, our findings simplify, improve, and extend previous results by Bienstock and Zuckerberg on covering problems.
AbstractList	In convex integer programming, various procedures have been developed to strengthen convex relaxations of sets of integer points. On the one hand, there exist several general-purpose methods that strengthen relaxations without specific knowledge of the set S of feasible integer points, such as popular linear programming or semi-definite programming hierarchies. On the other hand, various methods have been designed for obtaining strengthened relaxations for very specific sets S that arise in combinatorial optimization. We propose a new efficient method that interpolates between these two approaches. Our procedure strengthens any convex set containing a set S ⊆ { 0 , 1 } n by exploiting certain additional information about S. Namely, the required extra information will be in the form of a Boolean formula ϕ defining the target set S. The new relaxation is obtained by "feeding" the convex set into the formula ϕ . We analyze various aspects regarding the strength of our procedure. As one application, interpreting an iterated application of our procedure as a hierarchy, our findings simplify, improve, and extend previous results by Bienstock and Zuckerberg on covering problems.In convex integer programming, various procedures have been developed to strengthen convex relaxations of sets of integer points. On the one hand, there exist several general-purpose methods that strengthen relaxations without specific knowledge of the set S of feasible integer points, such as popular linear programming or semi-definite programming hierarchies. On the other hand, various methods have been designed for obtaining strengthened relaxations for very specific sets S that arise in combinatorial optimization. We propose a new efficient method that interpolates between these two approaches. Our procedure strengthens any convex set containing a set S ⊆ { 0 , 1 } n by exploiting certain additional information about S. Namely, the required extra information will be in the form of a Boolean formula ϕ defining the target set S. The new relaxation is obtained by "feeding" the convex set into the formula ϕ . We analyze various aspects regarding the strength of our procedure. As one application, interpreting an iterated application of our procedure as a hierarchy, our findings simplify, improve, and extend previous results by Bienstock and Zuckerberg on covering problems. In convex integer programming, various procedures have been developed to strengthen convex relaxations of sets of integer points. On the one hand, there exist several general-purpose methods that strengthen relaxations without specific knowledge of the set S of feasible integer points, such as popular linear programming or semi-definite programming hierarchies. On the other hand, various methods have been designed for obtaining strengthened relaxations for very specific sets S that arise in combinatorial optimization. We propose a new efficient method that interpolates between these two approaches. Our procedure strengthens any convex set containing a set $$ S \subseteq \{0,1\}^n $$ S⊆0,1n by exploiting certain additional information about S. Namely, the required extra information will be in the form of a Boolean formula $$\phi $$ ϕ defining the target set S. The new relaxation is obtained by “feeding” the convex set into the formula $$\phi $$ ϕ. We analyze various aspects regarding the strength of our procedure. As one application, interpreting an iterated application of our procedure as a hierarchy, our findings simplify, improve, and extend previous results by Bienstock and Zuckerberg on covering problems. In convex integer programming, various procedures have been developed to strengthen convex relaxations of sets of integer points. On the one hand, there exist several general-purpose methods that strengthen relaxations without specific knowledge of the set S of feasible integer points, such as popular linear programming or semi-definite programming hierarchies. On the other hand, various methods have been designed for obtaining strengthened relaxations for very specific sets S that arise in combinatorial optimization. We propose a new efficient method that interpolates between these two approaches. Our procedure strengthens any convex set containing a set S⊆{0,1}n by exploiting certain additional information about S. Namely, the required extra information will be in the form of a Boolean formula ϕ defining the target set S. The new relaxation is obtained by “feeding” the convex set into the formula ϕ. We analyze various aspects regarding the strength of our procedure. As one application, interpreting an iterated application of our procedure as a hierarchy, our findings simplify, improve, and extend previous results by Bienstock and Zuckerberg on covering problems. In convex integer programming, various procedures have been developed to strengthen convex relaxations of sets of integer points. On the one hand, there exist several general-purpose methods that strengthen relaxations without specific knowledge of the set S of feasible integer points, such as popular linear programming or semi-definite programming hierarchies. On the other hand, various methods have been designed for obtaining strengthened relaxations for very specific sets S that arise in combinatorial optimization. We propose a new efficient method that interpolates between these two approaches. Our procedure strengthens any convex set containing a set S ⊆ { 0 , 1 } n by exploiting certain additional information about S . Namely, the required extra information will be in the form of a Boolean formula ϕ defining the target set S . The new relaxation is obtained by “feeding” the convex set into the formula ϕ . We analyze various aspects regarding the strength of our procedure. As one application, interpreting an iterated application of our procedure as a hierarchy, our findings simplify, improve, and extend previous results by Bienstock and Zuckerberg on covering problems. In convex integer programming, various procedures have been developed to strengthen convex relaxations of sets of integer points. On the one hand, there exist several general-purpose methods that strengthen relaxations without specific knowledge of the set S of feasible integer points, such as popular linear programming or semi-definite programming hierarchies. On the other hand, various methods have been designed for obtaining strengthened relaxations for very specific sets S that arise in combinatorial optimization. We propose a new efficient method that interpolates between these two approaches. Our procedure strengthens any convex set containing a set $$ S \subseteq \{0,1\}^n $$ S ⊆ { 0 , 1 } n by exploiting certain additional information about S . Namely, the required extra information will be in the form of a Boolean formula $$\phi $$ ϕ defining the target set S . The new relaxation is obtained by “feeding” the convex set into the formula $$\phi $$ ϕ . We analyze various aspects regarding the strength of our procedure. As one application, interpreting an iterated application of our procedure as a hierarchy, our findings simplify, improve, and extend previous results by Bienstock and Zuckerberg on covering problems.
Author	Huynh, Tony Fiorini, Samuel Weltge, Stefan
Author_xml	– sequence: 1 givenname: Samuel surname: Fiorini fullname: Fiorini, Samuel organization: Université libre de Bruxelles – sequence: 2 givenname: Tony surname: Huynh fullname: Huynh, Tony organization: Monash University – sequence: 3 givenname: Stefan surname: Weltge fullname: Weltge, Stefan email: weltge@tum.de organization: Technical University of Munich
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Cites_doi	10.1007/978-3-319-33461-5_25 10.1007/978-3-319-59250-3_33 10.1287/moor.2017.0880 10.1016/j.ipl.2005.09.008 10.1109/FOCS.2015.73 10.1137/0403036 10.1137/0801013 10.1016/S0167-6377(98)00025-X 10.1007/978-3-642-15369-3_28 10.1007/11830924_38 10.1007/3-540-45535-3_23 10.1007/s10107-017-1134-7 10.1109/FOCS.2016.67 10.1145/3127497 10.1016/S0167-5060(08)70342-X 10.1016/0012-365X(73)90167-2 10.1007/s004930050057 10.1007/s10107-012-0574-3 10.1016/0020-0190(83)90011-X 10.1137/S1052623402420346 10.1145/2716307 10.1137/0403021 10.1137/1.9781611975031.53 10.1137/1.9781611974782.153 10.1145/146637.146684 10.1007/s10107-005-0598-z 10.1016/j.ipl.2012.02.009
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Snippet	In convex integer programming, various procedures have been developed to strengthen convex relaxations of sets of integer points. On the one hand, there exist...
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SubjectTerms	Boolean Boolean algebra Calculus of Variations and Optimal Control; Optimization Combinatorial analysis Combinatorics Full Length Paper Hierarchies Integer programming Linear programming Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Optimization Theoretical
Title	Strengthening convex relaxations of 0/1-sets using Boolean formulas
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