Tight Bounds for Online Graph Partitioning

Abstract

We consider the following online optimization problem. We are given a graph $G$ and each vertex of the graph is assigned to one of $\\ell$ servers, where servers have capacity $k$ and we assume that the graph has $\\ell \\cdot k$ vertices. Initially, $G$ does not contain any edges and then the edges of $G$ are revealed one-by-one. The goal is to design an online algorithm $\\operatorname{ONL}$, which always places the connected components induced by the revealed edges on the same server and never exceeds the server capacities by more than $\\varepsilon k$ for constant $\\varepsilon>0$. Whenever $\\operatorname{ONL}$ learns about a new edge, the algorithm is allowed to move vertices from one server to another. Its objective is to minimize the number of vertex moves. More specifically, $\\operatorname{ONL}$ should minimize the competitive ratio: the total cost $\\operatorname{ONL}$ incurs compared to an optimal offline algorithm $\\operatorname{OPT}$. \nOur main contribution is a polynomial-time randomized algorithm, that is asymptotically optimal: we derive an upper bound of $O(\\log \\ell + \\log k)$ on its competitive ratio and show that no randomized online algorithm can achieve a competitive ratio of less than $\\Omega(\\log \\ell + \\log k)$. We also settle the open problem of the achievable competitive ratio by deterministic online algorithms, by deriving a competitive ratio of $\\Theta(\\ell \\lg k)$; to this end, we present an improved lower bound as well as a deterministic polynomial-time online algorithm. \nOur algorithms rely on a novel technique which combines efficient integer programming with a combinatorial approach for maintaining ILP solutions. We believe this technique is of independent interest and will find further applications in the future.