Theory of linear and integer programming
Theory of linear and integer programming
A lift-and-project cutting plane algorithm for mixed 0-1 programs
Mathematical Programming: Series A and B
A lifting procedure for the asymmetric traveling salesman polytope and a large new class of facets
Mathematical Programming: Series A and B
The graphical relaxation: a new framework for the Symmetric Traveling Salesman Polytope
Mathematical Programming: Series A and B
A course in computational algebraic number theory
A course in computational algebraic number theory
A new approach to the minimum cut problem
Journal of the ACM (JACM)
Mixed 0-1 programming by lift-and-project in a branch-and-cut framework
Management Science
Mathematical Programming: Series A and B
Separating Clique Tree and Bipartition Inequalities in Polynominal Time
Proceedings of the 4th International IPCO Conference on Integer Programming and Combinatorial Optimization
Combining and Strengthening Gomory Cuts
Proceedings of the 4th International IPCO Conference on Integer Programming and Combinatorial Optimization
Separating Maximally Violated Comb Inequalities in Planar Graphs
Proceedings of the 5th International IPCO Conference on Integer Programming and Combinatorial Optimization
Finding Cuts in the TSP (A preliminary report)
Finding Cuts in the TSP (A preliminary report)
Operations Research Letters
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Separation is of fundamental importance in cutting-plane based techniques for Integer Linear Programming (ILP). In recent decades, a considerable research effort has been devoted to the definition of effective separation procedures for families of well-structured cuts. In this paper we address the separation of Chvátal rank-1 inequalities in the context of general ILP's of the form min{cT x : Ax ≤ b, x integer}, where A is an m × n integer matrix and b an m-dimensional integer vector. In particular, for any given integer k we study mod-k cuts of the form λTAx ≤ ⌊λTb⌋ for any λ ∈ {0; 1/k,..., (k - 1)/k}m such that λTA is integer. Following the line of research recently proposed for mod- 2 cuts by Applegate, Bixby, Chvátal and Cook [1] and Fleischer and Tardos [16], we restrict to maximally violated cuts, i.e., to inequalities which are violated by (k - 1)/k by the given fractional point. We show that, for any given k, such a separation requires O(mn min {m,n}) time. Applications to the TSP are discussed. In particular, for any given k, we propose an O(|V|2|E*|)-time exact separation algorithm for mod-k cuts which are maximally violated by a given fractional TSP solution with support graph G* = (V,E*). This implies that we can identify a maximally violated TSP cut whenever a maximally violated (extended) comb inequality exists. Finally, specific classes of (sometimes new) facet-defining mod-k cuts for the TSP are analyzed.