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 knapsackcover 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 + epsilon)-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 + epsilon 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 + epsilon) factor.