Be Fair and Be Selfish! Characterizing Deterministic Diffusive Load-Balancing Schemes with Small Discrepancy

We consider the problem of deterministic distributed load balancing of indivisible tasks in the discrete setting. A set of n processors is connected into a d-regular symmetric network. In every time step, each processor exchanges some of the tasks allocated to it with each of their neighbours in the network. The goal is to minimize the discrepancy between the number of tasks on the most-loaded and the least-loaded processor as quickly as possible. In this model, the performance of load-balancing schemes obtained by rounding the continuous diffusion process up or down to the nearest integer was considered by Rabani et al. (1998), who showed that after T = O(log(Kn)/µ) steps any such scheme achieves a discrepancy of O(d log n/µ), where µ is the spectral gap of the transition matrix of the network graph, and K is the initial load discrepancy in the system. In this work, we identify natural additional conditions on the form of the discrete deterministic balancing scheme, which result in smaller values of discrepancy between maximum and minimum load. Specifically, we introduce the notion of a cumulatively fair load-balancing scheme, in which every node sends in total almost the same number of tasks along each of its outgoing edges during every interval of consecutive time steps, and not only at every single step. As our first main result, we prove that an