PTAS for Ordered Instances of Resource Allocation Problems

We consider the problem of fair allocation of indivisible goods where we are given a set I of m indivisible resources (items) and a set P of n customers (players) competing for the resources. Each resource j in I has a same value vj > 0 for a subset of customers interested in j and it has no value for other customers. The goal is to find a feasible allocation of the resources to the interested customers such that in the Max-Min scenario (also known as Santa Claus problem) the minimum utility (sum of the resources) received by each of the customers is as high as possible and in the Min-Max case (also known as R||C_max problem), the maximum utility is as low as possible. In this paper we are interested in instances of the problem that admit a PTAS. These instances are not only of theoretical interest but also have practical applications. For the Max-Min allocation problem, we start with instances of the problem that can be viewed as a convex bipartite graph; there exists an ordering of the resources such that each customer is interested (has positive evaluation) in a set of consecutive resources and we demonstrate a PTAS. For the Min-Max allocation problem, we obtain a PTAS for instances in which there is an ordering of the customers (machines) and each resource (job) is adjacent to a consecutive set of customers (machines). Next we show that our method for the Max-Min scenario, can be extended to a broader class of bipartite graphs where the resources can be viewed as a tree and each customer is interested in a sub-tree of a bounded number of leaves of this tree (e.g. a sub-path)

Scheduling data transfers in a network and the set scheduling problem

In this paper we consider the online ftp problem. The goal is to service a sequence of file transfer requests given bandwidth constraints of the underlying communication network. The main result of the paper is a technique that leads to algorithms that optimize several natural metrics, such as max-stretch, total flow time, max flow time, and total completion time. In particular, we show how to achieve optimum total flow time and optimum max-stretch if we increase the capacity of the underlying network by a logarithmic factor. We show that the resource augmentation is necessary by proving polynomial lower bounds on the max-stretch and total flow time for the case where online and offline algorithms are using same-capacity edges. Moreover, we also give poly-logarithmic lower bounds on the resource augmentation factor necessary in order to keep the total flow time and max-stretch within a constant factor of optimum

Randomized On-Line Scheduling of Parallel Jobs

We study randomized on-line scheduling on mesh machines. We show that for scheduling independent jobs randomized algorithms can achieve a significantly better performance than deterministic ones; on the other hand with dependencies randomization does not help. this research was done at Carnegie-Mellon University, Pittsburgh, PA, U.S.A. 1 Introduction In this paper we study the power of randomization for on-line scheduling of parallel jobs in a model which was introduced and studied in [4, 3] for deterministic scheduling. We give a randomized on-line algorithm for scheduling independent jobs on mesh machines which is significantly better than the optimal deterministic algorithm from [4]. On the other hand, for scheduling of jobs with dependencies on one-dimensional meshes, we show that randomization does not help---no randomized algorithm has asymptotically better performance on the worst-case input than the optimal deterministic algorithm from [3]. These results are particularly i..