Simple lifted cover inequalities and hard knapsack problems

We consider a class of random knapsack instances described by Chvátal, who showed that with probability going to 1, such instances require an exponential number of branch-and-bound nodes. We show that even with the use of simple lifted cover inequalities, an exponential number of nodes is required with probability going to 1. It is not surprising that there exist integer programming (IP) instances for which solution by branch-and-bound requires an exponential number of nodes, since integer programming is an NP-complete problem while the linear programs solved at each branch-and-bound node are polynomially solvable. Examples of such instances were given by Jeroslow [5], who presented a set of simple instances of the knapsack problem which require an exponential number of branch-andbound nodes when branching on variables, and by Chvátal [1], who considered a class of random instances of the knapsack problem and showed that with probability converging to 1, such a random instance requires exponentially many branch-and-bound nodes to solve. Most modern IP solvers use branch-and-cut algorithms, which combine branchand-bound with the use of cutting planes. Gu, Nemhauser, and Savelsbergh [4] considered solving the knapsack problem with branch-and-cut. They presented a set of instances that require an exponential number of branch-and-bound nodes even with the addition of simple lifted cover inequalities. More recent work in proving exponential worst-case bounds in the presence of various cutting planes has been done by Dash [2], who proved worst-case exponential bounds in the presence of lift-and-project cuts, Chvátal-Gomory inequalities, and matrix cuts as described by Lovász and Schrijver. The work of Gu et al. and Dash is similar to Jeroslow’s work in that specific “worst-case” examples are presented. In this paper we build on Chvátal’s results, which are concerned with average-case performance over a class of random instances. We add all simple lifted cover inequalities to his formulation and show that an exponential number of branch-and-bound nodes is required with probability converging to 1. This result is not suggested by the NP-hardness of binary knapsack problems, because cover inequality separation for these problems is NP-hard [6].