Primal–Dual Algorithms for Precedence Constrained Covering Problems

Abstract

A covering problem is an integer linear program of type $\min \{c^Tx\mid Ax\ge D,\ 0\le x\le d,\ x \in \mathbb {Z}\}$ 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 (PCCPs) are of high theoretical and practical importance even in the special case of the precedence constrained knapsack problem, that is, 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 $\mathcal {P}$, that is, 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 $d\not \equiv 1$, we present pseudo-polynomial algorithms. Finally, we show that the problem does not admit a PTAS under standard complexity assumptions.