Partitioning a weighted partial order

The problem of partitioning a partially ordered set into a minimum number of chains is a well-known problem. In this paper we study a generalization of this problem, where we not only assume that the chains have bounded size, but also that a weight wi is given for each element i in the partial order such that wi ≤ wj if i ≺ j. The problem is then to partition the partial order into a minimum-weight set of chains of bounded size, where the weight of a chain equals the weight of the heaviest element in the chain. We prove that this problem is APX -hard, and we propose and analyze lower bounds for this problem. Based on these lower bounds, we exhibit a 2-approximation algorithm, and show that it is tight. We report computational results for a number of real-world and randomly generated problem instances.