Tight Bounds for Online Graph Partitioning
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We consider the following online optimization problem. We are given a graph G and each vertex of the graph is assigned to one of ℓ servers, where servers have capacity k and we assume that the graph has ℓ· 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 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 ε k for constant ε>0. Whenever 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, ONL should minimize the competitive ratio: the total cost ONL incurs compared to an optimal offline algorithm OPT. Our main contribution is a polynomial-time randomized algorithm, that is asymptotically optimal: we derive an upper bound of O(logℓ + log k) on its competitive ratio and show that no randomized online algorithm can achieve a competitive ratio of less than Ω(logℓ + log k). We also settle the open problem of the achievable competitive ratio by deterministic online algorithms, by deriving a competitive ratio of Θ(ℓ k); to this end, we present an improved lower bound as well as a deterministic polynomial-time online algorithm. Our 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.
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