Task Assignment in Tree-Like Hierarchical Structures

Abstract

Most large organizations, such as corporations, are hierarchical organizations. In hierarchical organizations each entity in the organization, except the root entity, is a sub-part of another entity. In this paper we study the task assignment problem to the entities of tree-like hierarchical organizations. The inherent tree structure introduces an interesting and challenging constraint to the standard assignment problem. When a task is assigned to an entity in a hierarchical organization, the whole entity, including its sub-entities, is responsible from the execution of that particular task. In other words, if an entity has been assigned to a task, neither its descendants nor its ancestors can be assigned to a task. Sub-entities cannot be assigned as they have an ancestor already occupied. Ancestor entities cannot be assigned since one of their sub-entities has already been employed in an assignment. In the paper, we formally introduce this new version of the assignment problem called Maximum Weight Tree Matching ($MWTM$), and show its NP-hardness. We also propose an effective heuristic solution based on an iterative LP-relaxation to it.