The resource dependent assignment problem with a convex agent cost function

We study the resource dependent assignment problem (RDAP), for which the cost of assigning agent j to task i is a multiplication of task i's cost parameter by a cost function of agent j, which is a convex function of the amount of resource allocated for the agent to execute his task. The quality of a solution is measured by two criteria. The first is the total assignment cost, and the second is the total weighted resource consumption. We consider four different variations of the RDAP and prove that three are NP-hard, while the last is polynomially solvable under some acceptable assumptions. In our NP-hardness proof we use a very general instance which makes the proof applicable to a large set of special cases of the RDAP, including several important scheduling problems whose complexity was unresolved heretofore. In addition, we design a novel approximation algorithm for one of the NP-hard variations of the RDAP with a very tight approximation ratio for any practical problem found in the literature.