Polyhedral Results for the Precedence-Constrained Knapsack Problem

Abstract A problem characteristic common to a number of important integer programming problems is that of precedence constraints: a transitive collection of constraints of the form xj ≤ xi with 0 ≤ xi ≤ 1, 0 ≤ xj ≤ 1, xi, xj integer. Precedence constraints are of interest both because they arise frequently in integer programming applications and because the convex hull of feasible integer points is the same as the region obtained by relaxing the integrality restrictions. This paper investigates the polyhedral structure of the convex hull of feasible integer points when the precedence constraints are complicated by an additional constraint or, more generally, by additional constraints defining an independence system. Sequential lifting for independence systems with precedence constraints is discussed and extensions of the cover and 1-configuration inequalities known for the knapsack polytope are presented. A general procedure for inducing facets called rooting is introduced and a variety of facets based on rooting are developed. The paper concludes by discussing a procedure for coalescing constraints related to minimal covers into facets.

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