Efficient schemes for nearest neighbor load balancing

We design a general mathematical framework to analyze the properties of nearest neighbor balancing algorithms of the di€usion type. Within this framework we develop a new Optimal Polynomial Scheme (OPS) which we show to terminate within a ®nite number m of steps, where m only depends on the graph and not on the initial load distribution. We show that all existing di€usion load balancing algorithms, including OPS, determine a ̄ow of load on the edges of the graph which is uniquely de®ned, independent of the method and minimal in the l2-norm. This result can also be extended to edge weighted graphs. The l2-minimality is achieved only if a di€usion algorithm is used as preprocessing and the real movement of load is performed in a second step. Thus, it is advisable to split the balancing process into the two steps of ®rst determining a balancing ̄ow and afterwards moving the load. We introduce the problem of scheduling a ̄ow and present some ®rst results on its complexity and the approximation quality of local greedy heuristics. Ó 1999 Published by Elsevier Science B.V. All rights reserved.