Complexity analysis of an assignment problem with controllable assignment costs and its applications in scheduling

We extend the classical linear assignment problem to the case where 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. The cost function of agent j is a linear function of the amount of resource allocated to the agent. A solution for our assignment problem is de…ned by the assignment of agents to tasks and by a resource allocation to each agent. The quality of a solution is measured by two criteria. The …rst criterion is the total assignment cost and the second is the total weighted resource consumption. We address these criteria via four di¤erent problem variations. We prove that our assignment problem is NP-hard for three of the four variations even if all the resource consumption weights are equal. However, and somewhat surprisingly, we …nd that the fourth variation is solvable in polynomial time. In addition, we …nd that our assignment problem is equivalent to a large set of important scheduling problems whose complexity has heretofore been an open question for three of the four variations. Keywords: Assignment problem, scheduling, controllable processing times, complexity, resource allocation, bicriteria optimization. 1 Introduction This paper presents and analyzes a new extension of the classical linear assignment problem (LAP) which has some very important applications in deterministic scheduling theory. Assignment problems deal with the question of how to assign a set of n agents to a set of n tasks such that each task is performed only once and each agent is assigned to a single task such that a speci…c prede…ned objective will be minimized. An This research was supported by THE ISRAEL SCIENCE FOUNDATION (grant No. 633/08). Partial support by the Paul Ivanier Center for Robotics and Production Management, Ben-Gurion University of the Negev is also gratefully acknowledged. ye-mails: [email protected] ze-mails: [email protected] and [email protected]