Theory of linear and integer programming
Theory of linear and integer programming
Minkowski's convex body theorem and integer programming
Mathematics of Operations Research
A hierarchy of polynomial time lattice basis reduction algorithms
Theoretical Computer Science
Chva´tal closures for mixed integer programming problems
Mathematical Programming: Series A and B
The generalized basis reduction algorithm
Mathematics of Operations Research
A lift-and-project cutting plane algorithm for mixed 0-1 programs
Mathematical Programming: Series A and B
A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed
Mathematics of Operations Research
On Barvinok's algorithm for counting lattice points in fixed dimension
Mathematics of Operations Research
Solving a System of Linear Diophantine Equations with Lower and Upper Bounds on the Variables
Mathematics of Operations Research
Decomposition of Integer Programs and of Generating Sets
ESA '97 Proceedings of the 5th Annual European Symposium on Algorithms
A Class of Hard Small 0-1 Programs
INFORMS Journal on Computing
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
Improved algorithms for integer programming and related lattice problems
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
Hard Equality Constrained Integer Knapsacks
Mathematics of Operations Research
Basis reduction and the complexity of branch-and-bound
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
On the Complexity of Selecting Disjunctions in Integer Programming
SIAM Journal on Optimization
Journal of Global Optimization
Analysis of integer programming algorithms with L-partition and unimodular transformations
Automation and Remote Control
Bounds on the size of branch-and-bound proofs for integer knapsacks
Operations Research Letters
On the structure of reduced kernel lattice bases
IPCO'13 Proceedings of the 16th international conference on Integer Programming and Combinatorial Optimization
The Cunningham-Geelen Method in Practice: Branch-Decompositions and Integer Programming
INFORMS Journal on Computing
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We propose a very simple preconditioning method for integer programming feasibility problems: replacing the problem b^'@?Ax@?bx@?Z^n with b^'@?(AU)y@?by@?Z^n, where U is a unimodular matrix computed via basis reduction, to make the columns of AU short (i.e. have small Euclidean norm), and nearly orthogonal (see e.g. [Arjen K. Lenstra, Hendrik W. Lenstra, Jr., Laszlo Lovasz, Factoring polynomials with rational coefficients, Mathematische Annalen 261 (1982) 515-534; Ravi Kannan, Minkowski's convex body theorem and integer programming, Mathematics of Operations Research 12 (3) (1987) 415-440]). Our approach is termed column basis reduction, and the reformulation is called rangespace reformulation. It is motivated by the technique proposed for equality constrained IPs by Aardal, Hurkens and Lenstra. We also propose a simplified method to compute their reformulation. We also study a family of IP instances, called decomposable knapsack problems (DKPs). DKPs generalize the instances proposed by Jeroslow, Chvatal and Todd, Avis, Aardal and Lenstra, and Cornuejols et al. They are knapsack problems with a constraint vector of the form pM+r, with p0 and r integral vectors, and M a large integer. If the parameters are suitably chosen in DKPs, we prove *hardness results, when branch-and-bound branching on individual variables is applied; *that they are easy, if one branches on the constraint px instead; and *that branching on the last few variables in either the rangespace or the AHL reformulations is equivalent to branching on px in the original problem. We also provide recipes to generate such instances. Our computational study confirms that the behavior of the studied instances in practice is as predicted by the theory.