Primal–Dual Algorithms for Precedence Constrained Covering Problems

A covering problem is an integer linear program of type min { c T x ∣ A x ≥ D , 0 ≤ x ≤ d , x ∈ Z } where A ∈ Z + m × n , D ∈ Z + m , and c , d ∈ Z + n . In this paper, we study covering problems with additional precedence constraints { x i ≤ x j ∀ j ⪯ i ∈ P } , where P = ( [ n ] , ⪯ ) 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 ≡ 1 . Our main result is a strongly-polynomial primal–dual approximation algorithm for PCCP with d ≡ 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 P , that is, by the size of a maximum antichain in 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 P . For the general case with d ≢ 1 , we present pseudo-polynomial algorithms. Finally, we show that the problem does not admit a PTAS under standard complexity assumptions.