
Convex Hull Formulations for MixedInteger Multilinear Functions
In this paper, we present convex hull formulations for a mixedinteger, ...
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MIP Formulations for the Steiner Forest Problem
The Steiner Forest problem is among the fundamental network design probl...
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Binary extended formulations and sequential convexification
A binarization of a bounded variable x is a linear formulation with vari...
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Knowledge engineering mixedinteger linear programming: constraint typology
In this paper, we investigate the constraint typology of mixedinteger l...
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Compact Mathematical Programs For DECMDPs With Structured Agent Interactions
To deal with the prohibitive complexity of calculating policies in Decen...
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Generating Target Graph Couplings for QAOA from Native Quantum Hardware Couplings
We present methods for constructing any target coupling graph using limi...
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When LiftandProject Cuts are Different
In this paper, we present a method to determine if a liftandproject cu...
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Lifting Linear Extension Complexity Bounds to the MixedInteger Setting
Mixedinteger mathematical programs are among the most commonly used models for a wide set of problems in Operations Research and related fields. However, there is still very little known about what can be expressed by small mixedinteger programs. In particular, prior to this work, it was open whether some classical problems, like the minimum oddcut problem, can be expressed by a compact mixedinteger program with few (even constantly many) integer variables. This is in stark contrast to linear formulations, where recent breakthroughs in the field of extended formulations have shown that many polytopes associated to classical combinatorial optimization problems do not even admit approximate extended formulations of subexponential size. We provide a general framework for lifting inapproximability results of extended formulations to the setting of mixedinteger extended formulations, and obtain almost tight lower bounds on the number of integer variables needed to describe a variety of classical combinatorial optimization problems. Among the implications we obtain, we show that any mixedinteger extended formulation of subexponential size for the matching polytope, cut polytope, traveling salesman polytope or dominant of the oddcut polytope, needs Ω(n/ n) many integer variables, where n is the number of vertices of the underlying graph. Conversely, the abovementioned polyhedra admit polynomialsize mixedinteger formulations with only O(n) or O(n n) (for the traveling salesman polytope) many integer variables. Our results build upon a new decomposition technique that, for any convex set C , allows for approximating any mixedinteger description of C by the intersection of C with the union of a small number of affine subspaces.
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