Separation Algorithms for 0-1 Knapsack Polytopes

Konstantinos Kaparis (k.kaparissoton.ac.uk)
Adam Letchford (A.N.Letchfordlancaster.ac.uk)

Abstract: Valid inequalities for 0-1 knapsack polytopes often prove useful when tackling hard 0-1 Linear Programming problems. To use such inequalities effectively, one needs separation algorithms for them, i.e., routines for detecting when they are violated. We show that the separation problems for the so-called extended cover and weight inequalities can be solved exactly in O(nb) time and O((n+a_{max})b) time, respectively, where n is the number of items, b is the knapsack capacity and a_{max} is the largest item weight. We also present fast and effective separation heuristics for the extended cover and lifted cover inequalities. Finally, we present a new exact separation algorithm for the 0-1 knapsack polytope itself, which is faster than existing methods. Extensive computational results are also given.

Keywords: integer programming, branch-and-cut, knapsack problems

Category 1: Integer Programming (0-1 Programming )

Category 2: Integer Programming (Cutting Plane Approaches )

Category 3: Combinatorial Optimization (Polyhedra )

Citation: Working paper, Department of Management Science, Lancaster University, June 2007. Revised February 2008.

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Entry Submitted: 06/07/2007
Entry Accepted: 06/07/2007
Entry Last Modified: 03/14/2008

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