cient 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 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. 2