Tight discrepancy bounds for discrete iterative load balancing within the continuous spectral time

Determine a tight bound on the discrepancy achieved by the discrete iterative load balancing model on an undirected, connected graph G=(V,E), starting from an arbitrary integral initial load vector with initial discrepancy K, after O(tau_cont(K)) rounds, where tau_cont(K) is the number of rounds the corresponding continuous (fractional) load balancing process needs to reach discrepancy at most 1.

Background

The paper studies iterative load balancing on graphs with discrete, indivisible tokens, contrasting it with the well-understood continuous (fractional) setting where Markov chain spectral properties provide tight convergence rates. Discrete rounding introduces non-linearity and dependencies that complicate analysis, and a long-standing goal has been to quantify the precise relationship between the continuous and discrete settings.

The authors reference a concrete formulation originating from prior work that asks for tight discrepancy bounds in the discrete model within the time scale dictated by the continuous spectral analysis. Their results nearly close this gap by proving that a discrepancy of 3 can be reached, with rounds matching the continuous spectral bound across several matching-based models, but they note this core question in its general form remains central.