Efficient schemes for nearest neighbor load balancing

We design a general mathematical framework to analyze the properties of nearest neighbor balancing algorithms of the diffusion type. Within this framework we develop a new Optimal Polynomial Scheme (OPS) which we show to terminate within a finite number m of steps, where m only depends on the graph and not on the initial load distribution. We show that all existing diffusion load balancing algorithms, including OPS, determine a flow of load on the edges of the graph which is uniquely defined, independent of the method and minimal in the l(2)-norm. This result can also be extended to edge weighted graphs. The l(2)-minimality is achieved only if a diffusion 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 first determining a balancing flow and afterwards moving the load. We introduce the problem of scheduling a flow and present some first results on its complexity and the approximation quality of local greedy heuristics. (C) 1999 Published by Elsevier Science B.V. All rights reserved.