A hierarchy of relaxation between the continuous and convex hull representations
SIAM Journal on Discrete Mathematics
A lift-and-project cutting plane algorithm for mixed 0-1 programs
Mathematical Programming: Series A and B
The Steiner tree problem I: formulations, compositions and extension of facets
Mathematical Programming: Series A and B
The Steiner tree problem II: properties and classes of facets
Mathematical Programming: Series A and B
Hilbert bases and the facets of special knapsack polytopes
Mathematics of Operations Research
Solving a System of Linear Diophantine Equations with Lower and Upper Bounds on the Variables
Mathematics of Operations Research
Market Split and Basis Reduction: Towards a Solution of the Cornuéjols-Dawande Instances
INFORMS Journal on Computing
Combining Problem Structure with Basis Reduction to Solve a Class of Hard Integer Programs
Mathematics of Operations Research
Improving Discrete Model Representations via Symmetry Considerations
Management Science
A Comparison of the Sherali-Adams, Lovász-Schrijver, and Lasserre Relaxations for 0--1 Programming
Mathematics of Operations Research
An algebraic approach to symmetric extended formulations
ISCO'12 Proceedings of the Second international conference on Combinatorial Optimization
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We introduce a general technique for creating an extended formulation of a mixed-integer program. We classify the integer variables into blocks, each of which generates a finite set of vector values. The extended formulation is constructed by creating a new binary variable for each generated value. Initial experiments show that the extended formulation can have a more compact complete description than the original formulation. We prove that, using this reformulation technique, the facet description decomposes into one ''linking polyhedron'' per block and the ''aggregated polyhedron''. Each of these polyhedra can be analyzed separately. For the case of identical coefficients in a block, we provide a complete description of the linking polyhedron and a polynomial-time separation algorithm. Applied to the knapsack with a fixed number of distinct coefficients, this theorem provides a complete description in an extended space with a polynomial number of variables. On the basis of this theory, we propose a new branching scheme that analyzes the problem structure. It is designed to be applied in those subproblems of hard integer programs where LP-based techniques do not provide good branching decisions. Preliminary computational experiments show that it is successful for some benchmark problems of multi-knapsack type.