Popular Conjectures as a Barrier for Dynamic Planar Graph Algorithms

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. The best known algorithm performs queries and updates in Õ(n2/3) time, based on ideas of a seminal paper by Fakcharoenphol and Rao [FOCS'01]. A (1+ε)-approximation algorithm of Abraham et al. [STOC'12] performs updates and queries in Õ(√n) time. An algorithm with a more practical O(polylog(n)) runtime would be a major breakthrough. However, such runtimes are only known for a (1+ε)-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(n1/2-ε) time, for any ε>0, unless the classical all-pairs-shortest-paths problem can be solved in truly subcubic time, which is widely believed to be impossible. We extend these results to obtain strong lower bounds for other related problems as well as for possible trade-offs between query and update time. Interestingly, our lower bounds hold even in very restrictive models where only weight updates are allowed.

[1]  David Peleg,et al.  Dynamic (1 + ∊)-Approximate Matchings: A Density-Sensitive Approach , 2016, SODA.

[2]  Andrew V. Goldberg,et al.  On Dynamic Approximate Shortest Paths for Planar Graphs with Worst-Case Costs , 2016, SODA.

[3]  Avi Wigderson,et al.  Proceedings of the twenty-fifth annual ACM symposium on Theory of Computing , 1992, Symposium on the Theory of Computing.

[4]  Piotr Sankowski,et al.  Dynamic Plane Transitive Closure , 2007, ESA.

[5]  Amir Abboud,et al.  Popular Conjectures Imply Strong Lower Bounds for Dynamic Problems , 2014, 2014 IEEE 55th Annual Symposium on Foundations of Computer Science.

[6]  Jochen Messner,et al.  Planarizing Gadgets for Perfect Matching Do Not Exist , 2012, MFCS.

[7]  Liam Roditty,et al.  Fast approximation algorithms for the diameter and radius of sparse graphs , 2013, STOC '13.

[8]  Sandeep Sen,et al.  Fully Dynamic Maximal Matching in O (log n) Update Time , 2011, 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science.

[9]  Allan Grønlund Jørgensen,et al.  New Unconditional Hardness Results for Dynamic and Online Problems , 2015, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.

[10]  Giuseppe F. Italiano,et al.  Deterministic Fully Dynamic Data Structures for Vertex Cover and Matching , 2015, SODA.

[11]  Mihai Patrascu,et al.  Towards polynomial lower bounds for dynamic problems , 2010, STOC '10.

[12]  Virginia Vassilevska Williams,et al.  Multiplying matrices faster than coppersmith-winograd , 2012, STOC '12.

[13]  Philip N. Klein,et al.  Faster Shortest-Path Algorithms for Planar Graphs , 1997, J. Comput. Syst. Sci..

[14]  Avi Wigderson,et al.  Extractors and Rank Extractors for Polynomial Sources , 2007, FOCS.

[15]  Toniann Pitassi,et al.  Proceedings of the forty-fourth annual ACM symposium on Theory of computing , 2012, STOC 2012.

[16]  Uri Zwick,et al.  All-Pairs Almost Shortest Paths , 1997, SIAM J. Comput..

[17]  Piotr Sankowski,et al.  Single Source -- All Sinks Max Flows in Planar Digraphs , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[18]  Monika Henzinger,et al.  Unifying and Strengthening Hardness for Dynamic Problems via the Online Matrix-Vector Multiplication Conjecture , 2015, STOC.

[19]  Stephen Alstrup,et al.  Simpler, faster and shorter labels for distances in graphs , 2015, SODA.

[20]  Søren Dahlgaard,et al.  On the Hardness of Partially Dynamic Graph Problems and Connections to Diameter , 2016, ICALP.

[21]  Huacheng Yu,et al.  Matching Triangles and Basing Hardness on an Extremely Popular Conjecture , 2015, STOC.

[22]  Dorothea Wagner,et al.  Landmark-Based Routing in Dynamic Graphs , 2007, WEA.

[23]  Pawel Gawrychowski,et al.  Improved Bounds for Shortest Paths in Dense Distance Graphs , 2018, ICALP.

[24]  Philip N. Klein,et al.  Multiple-source shortest paths in planar graphs , 2005, SODA '05.

[25]  Philip N. Klein,et al.  A Fully Dynamic Approximation Scheme for Shortest Paths in Planar Graphs , 1998, Algorithmica.

[26]  Mathias Bæk Tejs Knudsen,et al.  Sublinear distance labeling for sparse graphs , 2015, ArXiv.

[27]  Sairam Subramanian.  A Fully Dynamic Data Structure for Reachability in Planar Digraphs , 1993, ESA.

[28]  Haim Kaplan,et al.  Submatrix maximum queries in Monge matrices and Monge partial matrices, and their applications , 2012, SODA.

[29]  Clifford Stein,et al.  Faster Fully Dynamic Matchings with Small Approximation Ratios , 2016, SODA.

[30]  Uri Zwick,et al.  On Dynamic Shortest Paths Problems , 2004, Algorithmica.

[31]  David Eppstein.  Dynamic Connectivity in Digital Images , 1997, Inf. Process. Lett..

[32]  Robert Krauthgamer,et al.  Orienting Fully Dynamic Graphs with Worst-Case Time Bounds , 2013, ICALP.

[33]  Peter Sanders,et al.  Engineering Route Planning Algorithms , 2009, Algorithmics of Large and Complex Networks.

[34]  Zvi Galil,et al.  Fully dynamic planarity testing (extended abstract) , 1992, STOC '92.

[35]  Fabrizio Grandoni,et al.  Subcubic Equivalences between Graph Centrality Problems, APSP, and Diameter , 2015, SODA.

[36]  Krzysztof Onak,et al.  Maintaining a large matching and a small vertex cover , 2010, STOC '10.

[37]  Raphael Yuster,et al.  Approximating the Diameter of Planar Graphs in Near Linear Time , 2011, TALG.

[38]  Piotr Sankowski,et al.  Faster dynamic matchings and vertex connectivity , 2007, SODA '07.

[39]  Leonard J. Schulman,et al.  Proceedings of the forty-second ACM symposium on Theory of computing , 2010, STOC 2010.

[40]  Seth Pettie,et al.  Linear-Time Approximation for Maximum Weight Matching , 2014, JACM.

[41]  Richard Peng,et al.  Fully Dynamic $(1+\epsilon)$-Approximate Matchings , 2013, 1304.0378.

[42]  Giuseppe F. Italiano.  Fully Dynamic Planarity Testing , 2008, Encyclopedia of Algorithms.

[43]  Dániel Marx,et al.  On the Optimality of Planar and Geometric Approximation Schemes , 2007, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07).

[44]  Shay Solomon,et al.  Simple deterministic algorithms for fully dynamic maximal matching , 2012, STOC '13.

[45]  Joshua R. Wang,et al.  Approximation and Fixed Parameter Subquadratic Algorithms for Radius and Diameter in Sparse Graphs , 2016, SODA.

[46]  Amir Abboud,et al.  Exact Weight Subgraphs and the k-Sum Conjecture , 2013, ICALP.

[47]  Andrew V. Goldberg,et al.  Customizable Route Planning , 2011, SEA.

[48]  Haim Kaplan,et al.  A faster algorithm for the discrete Fréchet distance under translation , 2015, ArXiv.

[49]  Zvi Galil,et al.  Maintaining Biconnected Components of Dynamic Planar Graphs , 1991, ICALP.

[50]  Grammati E. Pantziou,et al.  Computing Shortest Paths and Distances in Planar Graphs , 1991, ICALP.

[51]  Tsvi Kopelowitz,et al.  Higher Lower Bounds from the 3SUM Conjecture , 2014, SODA.

[52]  Dániel Marx,et al.  The Square Root Phenomenon in Planar Graphs , 2013, ICALP.

[53]  Sandeep Sen,et al.  Maintaining Approximate Maximum Weighted Matching in Fully Dynamic Graphs , 2012, FSTTCS.

[54]  Kasper Green Larsen,et al.  Faster Online Matrix-Vector Multiplication , 2016, SODA.

[55]  François Le Gall,et al.  Powers of tensors and fast matrix multiplication , 2014, ISSAC.

[56]  Philip N. Klein,et al.  Faster Shortest-Path Algorithms for Planar Graphs , 1997, J. Comput. Syst. Sci..

[57]  Satish Rao,et al.  Planar graphs, negative weight edges, shortest paths, and near linear time , 2006, J. Comput. Syst. Sci..

[58]  Philip N. Klein,et al.  Multiple-Source Multiple-Sink Maximum Flow in Directed Planar Graphs in Near-Linear Time , 2011, FOCS.

[59]  Clifford Stein,et al.  Fully Dynamic Matching in Bipartite Graphs , 2015, ICALP.

[60]  Greg N. Frederickson,et al.  Data Structures for On-Line Updating of Minimum Spanning Trees, with Applications , 1985, SIAM J. Comput..

[61]  Barna Saha,et al.  Language Edit Distance and Maximum Likelihood Parsing of Stochastic Grammars: Faster Algorithms and Connection to Fundamental Graph Problems , 2014, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.

[62]  Ran Raz,et al.  Distance labeling in graphs , 2001, SODA '01.

[63]  Piotr Sankowski,et al.  Improved algorithms for min cut and max flow in undirected planar graphs , 2011, STOC '11.

[64]  Norbert Zeh,et al.  Orienting Dynamic Graphs, with Applications to Maximal Matchings and Adjacency Queries , 2014, ISAAC.

[65]  Peter Sanders,et al.  Dynamic Highway-Node Routing , 2007, WEA.

[66]  Peter Sanders,et al.  Exact Routing in Large Road Networks Using Contraction Hierarchies , 2012, Transp. Sci..

[67]  Ittai Abraham,et al.  Fully dynamic approximate distance oracles for planar graphs via forbidden-set distance labels , 2012, STOC '12.