Fast Dynamic Cuts, Distances and Effective Resistances via Vertex Sparsifiers

We present a general framework of designing efficient dynamic approximate algorithms for optimization problems on undirected graphs. In particular, we develop a technique that, given any problem that admits a certain notion of vertex sparsifiers, gives data structures that maintain approximate solutions in sub-linear update and query time. We illustrate the applicability of our paradigm to the following problems. (1)A fully-dynamic algorithm that approximates all-pair maximum-flows/minimum-cuts up to a nearly logarithmic factor in $\tilde{O}(n^{2/3})$ 11The $\tilde{O}(\cdot)$ notation is used in this paper to hide poly-logarithmic factors. amortized time against an oblivious adversary, and $\tilde{O}(m^{3/4})$ time against an adaptive adversary. (2)An incremental data structure that maintains $O(1)$ - approximate shortest path in $n^{o(1)}$ time per operation, as well as fully dynamic approximate all-pair shortest path and transshipment in $\tilde{O}(n^{2/3 +o(1)})$ amortized time per operation. (3)A fully-dynamic algorithm that approximates all-pair effective resistance up to an ($1+\epsilon$) factor in $\tilde{O}(n^{2/3+o(1)}\epsilon^{-O(1)})$ amortized update time per operation. The key tool behind result (1) is the dynamic maintenance of an algorithmic construction due to Madry [FOCS' 10], which partitions a graph into a collection of simpler graph structures (known as $j$-trees) and approximately captures the cut-flow and metric structure of the graph. The $O(1)$-approximation guarantee of (2) is by adapting the distance oracles by [Thorup-Zwick JACM '05]. Result (3) is obtained by invoking the random-walk based spectral vertex sparsifier by [Durfee et al. STOC '19] in a hierarchical manner, while carefully keeping track of the recourse among levels in the hierarchy. See https://arxiv.org/pdf/2005.02368.pdf for the full version of this paper.

[1]  Jure Leskovec,et al.  Motifs in Temporal Networks , 2016, WSDM.

[2]  Boaz Patt-Shamir,et al.  Near-Optimal Distributed Maximum Flow , 2015, SIAM J. Comput..

[3]  Konstantin Makarychev,et al.  Metric extension operators, vertex sparsifiers and Lipschitz extendability , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[4]  Mikkel Thorup,et al.  Fully-dynamic min-cut , 2001, STOC '01.

[5]  Richard Peng,et al.  On Fully Dynamic Graph Sparsifiers , 2016, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

[6]  Mikkel Thorup,et al.  Approximate distance oracles , 2001, JACM.

[7]  Mikkel Thorup,et al.  Deterministic Constructions of Approximate Distance Oracles and Spanners , 2005, ICALP.

[8]  Mikkel Thorup,et al.  Worst-case update times for fully-dynamic all-pairs shortest paths , 2005, STOC '05.

[9]  Frank Thomson Leighton,et al.  Vertex Sparsifiers and Abstract Rounding Algorithms , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[10]  Thatchaphol Saranurak,et al.  The Expander Hierarchy and its Applications to Dynamic Graph Algorithms , 2020, SODA.

[11]  Mikkel Thorup,et al.  Incremental Exact Min-Cut in Polylogarithmic Amortized Update Time , 2018, ACM Trans. Algorithms.

[12]  Marvin Künnemann,et al.  Fréchet Distance Under Translation: Conditional Hardness and an Algorithm via Offline Dynamic Grid Reachability , 2019, SODA.

[13]  Mikkel Thorup,et al.  Faster Algorithms for Edge Connectivity via Random 2-Out Contractions , 2019, SODA.

[14]  Claudio Gutierrez,et al.  Survey of graph database models , 2008, CSUR.

[15]  Xiaojin Zhu,et al.  --1 CONTENTS , 2006 .

[16]  Aleksander Madry,et al.  Fast Approximation Algorithms for Cut-Based Problems in Undirected Graphs , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[17]  Andrew V. Goldberg,et al.  Efficient maximum flow algorithms , 2014, CACM.

[18]  Jiawei Han,et al.  Data Mining: Concepts and Techniques , 2000 .

[19]  Shay Solomon,et al.  Local Algorithms for Bounded Degree Sparsifiers in Sparse Graphs , 2021, ITCS.

[20]  Gramoz Goranci,et al.  Dynamic Graph Algorithms and Graph Sparsification: New Techniques and Connections , 2019, ArXiv.

[21]  Nikhil Srivastava,et al.  Graph sparsification by effective resistances , 2008, SIAM J. Comput..

[22]  Ankur Moitra,et al.  Approximation Algorithms for Multicommodity-Type Problems with Guarantees Independent of the Graph Size , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[23]  Xiaorui Sun,et al.  Fully Dynamic c-Edge Connectivity in Subpolynomial Time , 2020, ArXiv.

[24]  Sebastian Krinninger,et al.  Dynamic low-stretch trees via dynamic low-diameter decompositions , 2018, STOC.

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

[26]  Shiri Chechik,et al.  Deterministic decremental single source shortest paths: beyond the o(mn) bound , 2016, STOC.

[27]  Doina Caragea,et al.  Graph Databases , 2019, Encyclopedia of Big Data Technologies.

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

[29]  Petra Perner,et al.  Data Mining - Concepts and Techniques , 2002, Künstliche Intell..

[30]  Stacy Patterson,et al.  Maximizing the Number of Spanning Trees in a Connected Graph , 2018, IEEE Transactions on Information Theory.

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

[32]  ThorupMikkel,et al.  Incremental Exact Min-Cut in Polylogarithmic Amortized Update Time , 2018 .

[33]  Christian Wulff-Nilsen,et al.  Fully-dynamic minimum spanning forest with improved worst-case update time , 2016, STOC.

[34]  Ittai Abraham,et al.  Nearly Tight Low Stretch Spanning Trees , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[35]  David Eppstein,et al.  Separator Based Sparsification. I. Planary Testing and Minimum Spanning Trees , 1996, J. Comput. Syst. Sci..

[36]  Uri Zwick,et al.  Improved Dynamic Reachability Algorithms for Directed Graphs , 2008, SIAM J. Comput..

[37]  Shang-Hua Teng,et al.  Spectral Sparsification of Graphs , 2008, SIAM J. Comput..

[38]  Peter G. Doyle,et al.  Random Walks and Electric Networks: REFERENCES , 1987 .

[39]  Shiri Chechik,et al.  Deterministic Partially Dynamic Single Source Shortest Paths for Sparse Graphs , 2017, SODA.

[40]  Shiri Chechik,et al.  Near-Optimal Approximate Decremental All Pairs Shortest Paths , 2018, 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS).

[41]  Olga Veksler,et al.  Fast approximate energy minimization via graph cuts , 2001, Proceedings of the Seventh IEEE International Conference on Computer Vision.

[42]  Lap Chi Lau,et al.  Fast matrix rank algorithms and applications , 2012, JACM.

[43]  Richard Peng,et al.  Optimal Offline Dynamic 2, 3-Edge/Vertex Connectivity , 2017, WADS.

[44]  Robert Krauthgamer,et al.  Preserving Terminal Distances Using Minors , 2014, SIAM J. Discret. Math..

[45]  David Eppstein,et al.  Separator-Based Sparsification II: Edge and Vertex Connectivity , 1999, SIAM J. Comput..

[46]  Uri Zwick,et al.  Dynamic approximate all-pairs shortest paths in undirected graphs , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[47]  Piotr Sankowski,et al.  Subquadratic Algorithm for Dynamic Shortest Distances , 2005, COCOON.

[48]  Robert Krauthgamer,et al.  Vertex Sparsifiers: New Results from Old Techniques , 2010, SIAM J. Comput..

[49]  Amir Abboud,et al.  Popular Conjectures as a Barrier for Dynamic Planar Graph Algorithms , 2016, 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS).

[50]  Ittai Abraham,et al.  Fully dynamic all-pairs shortest paths with worst-case update-time revisited , 2016, SODA.

[51]  Jonah Sherman,et al.  Area-convexity, l∞ regularization, and undirected multicommodity flow , 2017, STOC.

[52]  Richard Peng,et al.  Determinant-Preserving Sparsification of SDDM Matrices with Applications to Counting and Sampling Spanning Trees , 2017, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).

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

[54]  Hans-Peter Kriegel,et al.  Pattern Mining in Frequent Dynamic Subgraphs , 2006, Sixth International Conference on Data Mining (ICDM'06).

[55]  David Eppstein,et al.  Sparsification—a technique for speeding up dynamic graph algorithms , 1997, JACM.

[56]  Thatchaphol Saranurak,et al.  Dynamic spanning forest with worst-case update time: adaptive, Las Vegas, and O(n1/2 - ε)-time , 2017, STOC.

[57]  Christian Wulff-Nilsen,et al.  Dynamic Minimum Spanning Forest with Subpolynomial Worst-Case Update Time , 2017, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).

[58]  Frank Thomson Leighton,et al.  Extensions and limits to vertex sparsification , 2010, STOC '10.

[59]  Robert Krauthgamer,et al.  Refined Vertex Sparsifiers of Planar Graphs , 2020, SIAM J. Discret. Math..

[60]  Piotr Sankowski,et al.  Min-Cuts and Shortest Cycles in Planar Graphs in O(n loglogn) Time , 2011, ESA.

[61]  Richard Peng,et al.  Approximate Undirected Maximum Flows in O(mpolylog(n)) Time , 2014, SODA.

[62]  Jakub W. Pachocki,et al.  Solving SDD linear systems in nearly mlog1/2n time , 2014, STOC.

[63]  Richard Peng,et al.  Flows in almost linear time via adaptive preconditioning , 2019, STOC.

[64]  Thatchaphol Saranurak,et al.  Fully-Dynamic Graph Sparsifiers Against an Adaptive Adversary , 2020, ICALP.

[65]  Christian Schulz,et al.  Faster Fully Dynamic Transitive Closure in Practice , 2020, SEA.

[66]  Jonah Sherman,et al.  Breaking the Multicommodity Flow Barrier for O(vlog n)-Approximations to Sparsest Cut , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[67]  David Eppstein,et al.  Sparsification-a technique for speeding up dynamic graph algorithms , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[68]  Piotr Sankowski,et al.  The Power of Dynamic Distance Oracles: Efficient Dynamic Algorithms for the Steiner Tree , 2013, STOC.

[69]  Aaron Bernstein Maintaining Shortest Paths Under Deletions in Weighted Directed Graphs , 2016, SIAM J. Comput..

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

[71]  Thatchaphol Saranurak,et al.  Dynamic Matrix Inverse: Improved Algorithms and Matching Conditional Lower Bounds , 2019, 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS).

[72]  Piotr Sankowski,et al.  Dynamic Transitive Closure via Dynamic Matrix Inverse , 2004 .

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

[74]  Ittai Abraham,et al.  Fully Dynamic All-Pairs Shortest Paths: Breaking the O(n) Barrier , 2014, APPROX-RANDOM.

[75]  Yun Kuen Cheung,et al.  Graph Minors for Preserving Terminal Distances Approximately - Lower and Upper Bounds , 2016, ICALP.

[76]  Di Wang,et al.  Expander Decomposition and Pruning: Faster, Stronger, and Simpler , 2018, SODA.

[77]  Julia Chuzhoy On vertex sparsifiers with Steiner nodes , 2012, STOC '12.

[78]  Richard Peng,et al.  Fully dynamic spectral vertex sparsifiers and applications , 2019, STOC.

[79]  Richard Peng,et al.  Vertex Sparsifiers for c-Edge Connectivity , 2019, ArXiv.

[80]  David Eppstein,et al.  Offline Algorithms for Dynamic Minimum Spanning Tree Problems , 1991, J. Algorithms.

[81]  Harald Räcke,et al.  Optimal hierarchical decompositions for congestion minimization in networks , 2008, STOC.

[82]  Parinya Chalermsook,et al.  Mimicking Networks Parameterized by Connectivity , 2019, ArXiv.

[83]  Chintan Shah,et al.  Computing Cut-Based Hierarchical Decompositions in Almost Linear Time , 2014, SODA.

[84]  Richard Peng,et al.  An efficient parallel solver for SDD linear systems , 2013, STOC.

[85]  Manoj Gupta,et al.  Simple dynamic algorithms for Maximal Independent Set and other problems , 2018, ArXiv.

[86]  Ittai Abraham,et al.  Using petal-decompositions to build a low stretch spanning tree , 2012, STOC '12.

[87]  Pan Peng,et al.  The Power of Vertex Sparsifiers in Dynamic Graph Algorithms , 2017, ESA.

[88]  Liam Roditty,et al.  Improved dynamic algorithms for maintaining approximate shortest paths under deletions , 2011, SODA '11.

[89]  Piotr Sankowski,et al.  Dynamic transitive closure via dynamic matrix inverse: extended abstract , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[90]  Pan Peng,et al.  Dynamic Effective Resistances and Approximate Schur Complement on Separable Graphs , 2018, ESA.

[91]  Aaron Bernstein,et al.  Fully Dynamic (2 + epsilon) Approximate All-Pairs Shortest Paths with Fast Query and Close to Linear Update Time , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.