Dynamic Load Balancing under Restricting Updates

Abstract: We study the problem of load balancing in the dynamic setting. The input is a bipartite graph $G = (J \cup M, E)$ of jobs $J$ and machines $M$. Every job has a positive weight $w(j)$. The goal is to assign every job in $J$ to an adjacent machine in $M$ so as to minimize the maximum machine load, where the load of a machine is the sum of the weights of the jobs assigned to it. Although load balancing has been studied extensively in many models of computation, the only previous result in the dynamic model addresses the \emph{incremental} setting, where updates take the form of edge insertions and job deletions [BKS23b]. The authors show a $(1+ε)$-approximation for incremental \emph{fractional} load balancing with $\text{poly}(\log(n),1/ε)$ update time; they point out that their techniques do not extend to the decremental setting. Main Result: We obtain an algorithm which maintains a $(1+ε)$-approximate solution to the fractional load balancing problem under edge deletions and job insertions in $\tilde{O}(\log (n), 1/ε)$ update time. Our techniques are very different from those of the incremental algorithm of [BKS23b]. Instead, our algorithm is inspired by a load-balancing algorithm from the {\sc congest} model of computation [ALPZ21]. We develop a novel analytic framework for this algorithm, which allows us to show that it performs well in the dynamic setting. Rounding Scheme: We also present a black-box rounding scheme for dynamic load balancing. Plugging in our fractional algorithm, we obtain a decremental algorithm for $(2+ε)$-approximate \emph{integral} load balancing with $\text{poly}(\log(n),1/ε)$ update time. We also obtain the same bounds for incremental load balancing by plugging in the fractional algorithm of [BKS23b].