Popular Conjectures as a Barrier for Dynamic Planar Graph Algorithms

TL;DR

This paper establishes new conditional lower bounds for dynamic planar graph algorithms, showing that significantly faster algorithms for dynamic shortest paths and maximum weight bipartite matching are unlikely unless major conjectures in computational complexity are false.

Contribution

It introduces a novel framework for proving lower bounds in planar graphs based on popular conjectures, overcoming previous limitations due to non-planar reductions.

Findings

No dynamic planar shortest paths algorithm with both updates and queries in amortized O(n^{1/2 - ε}) time unless APSP is truly subcubic.

Similar lower bounds are shown for maximum weight bipartite matching in planar graphs.

These results connect the complexity of dynamic problems in planar graphs to fundamental conjectures in computational complexity.

Abstract

The dynamic shortest paths problem on planar graphs asks us to preprocess a planar graph $G$ such that we may support insertions and deletions of edges in $G$ as well as distance queries between any two nodes $u,v$ subject to the constraint that the graph remains planar at all times. This problem has been extensively studied in both the theory and experimental communities over the past decades and gets solved millions of times every day by companies like Google, Microsoft, and Uber. The best known algorithm performs queries and updates in $\tilde{O}(n^{2/3})$ time, based on ideas of a seminal paper by Fakcharoenphol and Rao [FOCS'01]. A $(1+\varepsilon)$-approximation algorithm of Abraham et al. [STOC'12] performs updates and queries in $\tilde{O}(\sqrt{n})$ time. An algorithm with $O(polylog(n))$ runtime would be a major breakthrough. However, such runtimes are only known for a $(1+\varepsilon)$-approximation in a model where only restricted weight updates are allowed due to Abraham et al. [SODA'16], or for easier problems like connectivity. In this paper, we follow a recent and very active line of work on showing lower bounds for polynomial time problems based on popular conjectures, obtaining the first such results for natural problems in planar graphs. Such results were previously out of reach due to the highly non-planar nature of known reductions and the impossibility of "planarizing gadgets". We introduce a new framework which is inspired by techniques from the literatures on distance labelling schemes and on parameterized complexity. Using our framework, we show that no algorithm for dynamic shortest paths or maximum weight bipartite matching in planar graphs can support both updates and queries in amortized $O(n^{\frac{1}{2}-\varepsilon})$ time, for $\varepsilon>0$, unless the classical APSP problem can be solved in truly subcubic time, [...]