Primal–Dual Algorithms for Precedence Constrained Covering Problems

Abstract

A covering problem is an integer linear program of type $$\min \{c^Tx\mid Ax\ge D,\ 0\le x\le d,\ x \in \mathbb {Z}\}$$min{cTxźAxźD,0≤x≤d,xźZ} where $$A\in \mathbb {Z}^{m\times n}_+$$AźZ+m×n, $$D\in \mathbb {Z}_+^m$$DźZ+m, and $$c,d\in \mathbb {Z}_+^n$$c,dźZ+n. In this paper, we study covering problems with additional precedence constraints $$\{x_i\le x_j \ \forall j\preceq i \in \mathcal {P}\}$${xi≤xjźjźiźP}, where $$\mathcal {P}=([n], \preceq )$$P=([n],ź) 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 $$m=1$$m=1 and $$d\equiv 1$$dź1. Our main result is a strongly-polynomial primal---dual approximation algorithm for PCCP with $$d\equiv 1$$dź1. 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 $$\mathcal {P}$$P, that is, by the size of a maximum antichain in $$\mathcal {P}$$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 $$\mathcal {P}$$P. For the general case with $$d\not \equiv 1$$dź1, we present pseudo-polynomial algorithms. Finally, we show that the problem does not admit a PTAS under standard complexity assumptions.