Primal-Dual Algorithms for Precedence Constrained Covering Problems

Abstract

AbstractA covering problem is an integer linear program of type \(\min \{c^Tx\mid Ax\ge D,\ 0\le x\le d,\ x \text{ integral }\}\) where \(A\in \mathbb {Z}^{m\times n}_+\), \(D\in \mathbb {Z}_+^m\), and \(c,d\in \mathbb {Z}_+^n\). In this paper, we study covering problems with additional precedence constraints \(\{x_i\le x_j \ \forall j\preceq i \in \mathcal {P}\}\), where \(\mathcal {P}=([n], \preceq )\) is some arbitrary, but fixed partial order on the items represented by the column-indices of \(A\). Such precedence constrained covering problems (PCCP) are of high theoretical and practical importance even in the special case of the precedence constrained knapsack problem, i.e., where \(m=1\) and \(d\equiv 1\).Our main result is a strongly-polynomial primal-dual approximation algorithm for PCCP with \(d\equiv 1\). Our approach generalizes the well-known knapsack cover inequalities to obtain an IP formulation which renders any explicit precedence constraints redundant. The approximation ratio of this algorithm is upper bounded by the width of \(\mathcal {P}\), i.e., by the size of a maximum antichain in \(\mathcal {P}\). Interestingly, this bound is independent of the number of constraints. We are not aware of any other results on approximation algorithms for PCCP on arbitrary posets \(\mathcal {P}\). For the general case with , we present pseudo-polynomial algorithms.KeywordsApproximation algorithmsPrecedence constraintsKnapsack problemCapacitated covering