Examples of new facets for the precedence constrained knapsack problem

We consider the polyhedral structure of the precedence constrained knapsack (PCK) problem, also known as the partially ordered knapsack problem. A set of items N is given, along with a partial order, or set of precedence relationships, on the items, denoted by S ⊆ N × N. A precedence relationship (i, j) ∈ S exists if item i can be placed in the knapsack only if item j is in the knapsack. Each item i ∈ N has a value ci ∈ Z and a weight ai ∈ Z, and the knapsack has a capacity b ∈ Z. The PCK problem seeks a maximum value subset of N whose total weight does not exceed the knapsack capacity, and that also satisfies the precedence relationships. The precedence constraints can be represented by the directed acyclic graph G = (N, S), where the node set is the set of all items N, and each precedence constraint in S is represented by a directed arc. Note that the precedence constraints are transitive, so without loss of generality we assume that S does not contain any redundant relationships, that is, S is the set of all immediate predecessor arcs. If G contains a cycle, all nodes within the cycle must either all be included in, or all be excluded from, the knapsack. Hence the cycle can be contracted into a single node, with cumulative value and weight coefficients, and the resulting directed graph is acyclic. An integer programming formulation of the PCK problem is as follows. Let