Abstract

Obtaining strong linear relaxations for capacitated covering problems constitutes a significant technical challenge. For one of the most basic cases, the relaxation based on knapsack-cover inequalities has an integrality gap of 2. We generalize the setting considering items that can be taken fractionally to cover a given demand, with a cost given by an arbitrary nondecreasing function (not necessarily convex) of the chosen fraction. We generalize the knapsack-cover inequalities and use them to obtain a polynomial (2 + \varepsilon )-approximation algorithm. Our primal-dual procedure has a natural interpretation as a water-filling algorithm, which overcomes the difficulties implied by having different growth rates in the cost functions: when the cost of an item increases slowly at some superior segment, it carefully increases the priority of all preceding segments. We generalize our algorithm to the Unsplittable Flow-Cover problem on a line, also for fractional items with nonlinear costs. We obtain a 4-approximation in pseudopolynomial time (4 + \varepsilon in polynomial time), matching the approximation ratio of the classical setting. We also present a rounding algorithm with an approximation guarantee of 2. This result is coupled with a polynomial time separation algorithm that allows solving our linear relaxation up to a loss of a (1 + \varepsilon ) factor.