PTAS for Ordered Instances of Resource Allocation Problems with Restrictions on Inclusions

We consider the problem of allocating a set I of m indivisible resources (items) to a set P of n customers (players) competing for the resources. Each resource j ∈ I has a same value vj > 0 for a subset of customers interested in j, and zero value for the remaining customers. The utility received by each customer is the sum of the values of the resources allocated to her. The goal is to find a feasible allocation of the resources to the interested customers such that for the Max-Min allocation problem (Min-Max allocation problem) the minimum of the utilities (maximum of the utilities) received by the customers is maximized (minimized). The Max-Min allocation problem is also known as the Fair Allocation problem, or the Santa Claus problem. The Min-Max allocation problem is the problem of Scheduling on Unrelated Parallel Machines, and is also known as the R | |Cmax problem. In this paper, we are interested in instances of the problem that admit a Polynomial Time Approximation Scheme (PTAS). We show that an ordering property on the resources and the customers is important and paves the way for a PTAS. For the Max-Min allocation problem, we start with instances of the problem that can be viewed as a convex bipartite graph; a bipartite graph for which there exists an ordering of the resources such that each customer is interested in (has a positive evaluation for) a set of consecutive resources. We demonstrate a PTAS for the inclusion-free cases. This class of instances is equivalent to the class of bipartite permutation graphs. For the Min-Max allocation problem, we also obtain a PTAS for inclusion-free instances. These instances are not only of theoretical interest but also have practical applications.