Improved Analysis of Online Balanced Clustering

Abstract

In the online balanced graph repartitioning problem, one has to maintain a clustering of n nodes into $\ell$ clusters, each having $k = n / \ell$ nodes. During runtime, an online algorithm is given a stream of communication requests between pairs of nodes: an inter-cluster communication costs one unit, while the intra-cluster communication is free. An algorithm can change the clustering, paying unit cost for each moved node.

This natural problem admits a simple $O(\ell ^2 \cdot k^2)$-competitive algorithm Comp, whose performance is far apart from the best known lower bound of $\varOmega (\ell \cdot k)$. One of open questions is whether the dependency on $\ell$ can be made linear; this question is of practical importance as in the typical datacenter application where virtual machines are clustered on physical servers, $\ell$ is of several orders of magnitude larger than k. We answer this question affirmatively, proving that a simple modification of Comp is $(\ell \cdot 2^{O(k)})$-competitive.

On the technical level, we achieve our bound by translating the problem to a system of linear integer equations and using Graver bases to show the existence of a “small” solution.

Supported by Polish National Science Centre grant 2016/22/E/ST6/00499, by Center for Foundations of Modern Computer Science (Charles University project UNCE/SCI/004), by the project 19-27871X of GA ČR, and by NSF CAREER grant 1651861.