Scheduling unsplittable flows using parallel switches

We address the problem of scheduling unsplittable flows using a number of switches in parallel. This has applications in optical switching and eliminates the need for re-sequencing in traditional packet switching. The use of parallel switches is becoming increasingly popular since it provides a way of building a high-speed switch while overcoming the speedup requirement imposed on the switch. Unlike packet switching however, we will assume that flows cannot be split across switches. This constraint adds a new dimension to the problem: various questions such as obtaining the best schedule, i.e. the schedule with the maximum throughput possible, become NP hard. Our problem is a special case of the general unsplittable flow problem, where in a directed capacitated graph containing a number of commodities with demands, the goal is to obtain a flow that does not violate capacity and in which all demands are satisfied and every commodity flows along a single path. In this paper, we are not going to address the general problem. Rather, we will study the special case of scheduling unsplittable flows using parallel switches, and present some simple approximation algorithms to various aspects of the problem with no speedup. We also define a speedup version of the problem and discuss under what speedup we can fully schedule an admissible set of flows.