Approximation algorithms for a bi-level knapsack problem

In this paper, we consider a variant of the knapsack problem. There are two knapsacks with probably different capacities, owned by two agents respectively. Given a set of items, each with a fixed size and a profit. The two agents select items and pack them into their own knapsacks under the capacity constraint. Same items can be packed simultaneously to different knapsacks. However, in this case the profit of such items may vary. One agent packs items into his knapsack to maximize the total profit, while another agent can only pack items into his knapsack as well but he cares about the total profits of items packed into two knapsacks. The latter agent is a leader while the former is a follower. We aim at designing an approximation algorithm for the leader assuming that the follower is selfish. For different settings we provide approximation results.