Near-Optimal Low-Congestion Shortcuts on Bounded Parameter Graphs

Abstract

We show that many distributed network optimization problems can be solved much more efficiently in structured and topologically simple networks.

It is known that solving essentially any global network optimization problem in a general network requires $\varOmega (\sqrt{n})$ rounds in the CONGEST model, even if the network diameter is small, e.g., logarithmic. Many networks of interest, however, have more structure which allows for significantly more efficient algorithms. Recently Ghaffari, Haeupler, Izumi and Zuzic [SODA’16,PODC’16] introduced low-congestion shortcuts as a suitable abstraction to capture this phenomenon. In particular, they showed that graphs with diameter D embeddable in a genus-g surface have good shortcuts and that these shortcuts lead to $\tilde{O}(g D)$-round algorithms for MST, Min-Cut and other problems.

We generalize these results by showing that networks with pathwidth or treewidth k allow for good shortcuts leading to fast $\tilde{O}(k D)$ distributed optimization algorithms. We also improve the dependence on genus g from $\tilde{O}(gD)$ to $\tilde{O}(\sqrt{g}D)$. Lastly, we prove lower bounds which show that the dependence on k and g in our shortcuts is optimal. Overall, this significantly refines and extends the understanding of how the complexity of distributed optimization problems depends on the network topology.

This work was supported in part by KAKENHI No. 15H00852 and 16H02878 as well as NSF grants CCF-1527110 and CCF-1618280.