Lifting valid inequalities for the precedence constrained knapsack problem

This paper considers the precedence constrained knapsack problem. More speci cally, we are interested in classes of valid inequalities which are facet-de ning for the precedence constrained knapsack polytope. We study the complexity of obtaining these facets using the standard sequential lifting procedure. Applying this procedure requires solving a combinatorial problem. For valid inequalities arising from minimal induced covers, we identify a class of lifting coe cients for which this problem can be solved in polynomial time, by using a supermodular function, and for which the values of the lifting coe cients have a combinatorial interpretation. For the remaining lifting coe cients it is shown that this optimization problem is strongly NP-hard. The same lifting procedure can be applied to (1,k)-con gurations, although in this case, the same combinatorial interpretation no longer applies. We also consider K-covers, to which the same procedure need not apply in general. We show that facets of the polytope can still be generated using a similar lifting technique. For tree knapsack problems, we observe that all lifting coe cients can be obtained in polynomial time. Computational experiments indicate that these facets signi cantly strengthen the LP-relaxation.