(Almost) Perfect Discrete Iterative Load Balancing

We consider discrete, iterative load balancing via matchings on arbitrary graphs. Initially each node holds a certain number of tokens, defining the load of the node, and the objective is to redistribute the tokens such that eventually each node has approximately the same number of tokens. We present results for a general class of simple local balancing schemes where the tokens are balanced via matchings. In each round the process averages the tokens of any two matched nodes. If the sum of their tokens is odd, the node to receive the one excess token is selected at random. Our class covers three popular models: in the matching model a new matching is generated randomly in each round, in the balancing circuit model a fixed sequence of matchings is applied periodically, and in the asynchronous model the load is balanced over a randomly chosen edge.We measure the quality of a load vector by its discrepancy, defined as the difference between the maximum and minimum load across all nodes. As our main result we show that with high probability our discrete balancing scheme reaches a discrepancy of 33 in a number of rounds which asymptotically matches the spectral bound for continuous load balancing with fractional load.This result improves and tightens a long line of previous works, by not only achieving a small constant discrepancy (instead of a non-explicit, large constant) but also holding for arbitrary instead of regular graphs. The result also demonstrates that in the general model we consider, discrete load balancing is no harder than continuous load balancing.