Tight bounds for online graph partitioning

Henzinger MH, Neumann S, Räcke H, Schmid S. 2021. Tight bounds for online graph partitioning. 32nd Annual ACM-SIAM Symposium on Discrete Algorithms. SODA: Symposium on Discrete Algorithms, 2799–2818.

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Author
Henzinger, MonikaISTA ; Neumann, Stefan; Räcke, Harald; Schmid, Stefan
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 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 ∊k for constant ∊ > 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. The problem was recently introduced by Henzinger et al. (SIGMETRICS'2019) and is related to classic online problems such as online paging and scheduling. It finds applications in the context of resource allocation in the cloud and for optimizing distributed data structures such as union–find data structures. Our main contribution is a polynomial-time randomized algorithm, that is asymptotically optimal: we derive an upper bound of O(log ℓ + log k) on its competitive ratio and show that no randomized online algorithm can achieve a competitive ratio of less than Ω(log ℓ + 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 log 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. More precisely, we use an ILP to assign the connected components induced by the revealed edges to the servers; this is similar to existing approximation schemes for scheduling algorithms. However, we cannot obtain our competitive ratios if we run the ILP after each edge insertion. Instead, we identify certain types of edge insertions, after which we can manually obtain an optimal ILP solution at zero cost without resolving the ILP. We believe this technique is of independent interest and will find further applications in the future.
Publishing Year
Date Published
2021-01-01
Proceedings Title
32nd Annual ACM-SIAM Symposium on Discrete Algorithms
Page
2799-2818
Conference
SODA: Symposium on Discrete Algorithms
Conference Location
Alexandria, VA, United States
Conference Date
2021-01-10 – 2021-01-13
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Cite this

Henzinger MH, Neumann S, Räcke H, Schmid S. Tight bounds for online graph partitioning. In: 32nd Annual ACM-SIAM Symposium on Discrete Algorithms. Society for Industrial and Applied Mathematics; 2021:2799-2818. doi:10.1137/1.9781611976465.166
Henzinger, M. H., Neumann, S., Räcke, H., & Schmid, S. (2021). Tight bounds for online graph partitioning. In 32nd Annual ACM-SIAM Symposium on Discrete Algorithms (pp. 2799–2818). Alexandria, VA, United States: Society for Industrial and Applied Mathematics. https://doi.org/10.1137/1.9781611976465.166
Henzinger, Monika H, Stefan Neumann, Harald Räcke, and Stefan Schmid. “Tight Bounds for Online Graph Partitioning.” In 32nd Annual ACM-SIAM Symposium on Discrete Algorithms, 2799–2818. Society for Industrial and Applied Mathematics, 2021. https://doi.org/10.1137/1.9781611976465.166.
M. H. Henzinger, S. Neumann, H. Räcke, and S. Schmid, “Tight bounds for online graph partitioning,” in 32nd Annual ACM-SIAM Symposium on Discrete Algorithms, Alexandria, VA, United States, 2021, pp. 2799–2818.
Henzinger MH, Neumann S, Räcke H, Schmid S. 2021. Tight bounds for online graph partitioning. 32nd Annual ACM-SIAM Symposium on Discrete Algorithms. SODA: Symposium on Discrete Algorithms, 2799–2818.
Henzinger, Monika H., et al. “Tight Bounds for Online Graph Partitioning.” 32nd Annual ACM-SIAM Symposium on Discrete Algorithms, Society for Industrial and Applied Mathematics, 2021, pp. 2799–818, doi:10.1137/1.9781611976465.166.
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