Approximating the Arboricity in Sublinear Time

We consider the problem of approximating the arboricity of a graph G “ pV ,Eq, which we denote by arbpGq, in sublinear time, where the arboricity of a graph is the minimal number of forests required to cover its edges. An algorithm for this problem may perform degree and neighbor queries, and is allowed a small error probability. We design an algorithm that outputs an estimate α̂, such that with probability 1 ́ 1{polypnq, arbpGq{c log n ď α̂ ď arbpGq, where n “ |V | and c is a constant. The expected query complexity and running time of the algorithm are Opn{arbpGqq ̈polyplognq, and this upper bound also holds with high probability. This bound is optimal for such an approximation up to a polyplognq factor. A conference version of this manuscript is to appear in SODA 2022. CSAIL at MIT, Boston University Department of Computer Science, talyaa01@gmail.com. Partially supported by the NSF Grant CCF-1740751, the Eric and Wendy Schmidt Fund, Ben-Gurion University, and the Computer Science Department at Boston University. CSAIL at MIT, saleet@mit.edu. Tel Aviv University, danaron@tau.ac.il. Partially supported by the Israel Science Foundation (grant No. 1041/18).

[1]  Hossein Jowhari,et al.  Tight bounds for Lp samplers, finding duplicates in streams, and related problems , 2010, PODS.

[2]  Dana Ron,et al.  The Arboricity Captures the Complexity of Sampling Edges , 2019, ICALP.

[3]  C. Nash-Williams.  Decomposition of Finite Graphs Into Forests , 1964 .

[4]  J. Edmonds.  Minimum partition of a matroid into independent subsets , 1965 .

[5]  Leizhen Cai,et al.  Random Separation: A New Method for Solving Fixed-Cardinality Optimization Problems , 2006, IWPEC.

[6]  Noga Alon,et al.  Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs , 2007, Algorithmica.

[7]  Marco Bressan.  Faster algorithms for counting subgraphs in sparse graphs , 2018 .

[8]  Maurice Queyranne,et al.  A network flow solution to some nonlinear 0-1 programming problems, with applications to graph theory , 1982, Networks.

[9]  Christoph Lenzen,et al.  Distributed minimum dominating set approximations in restricted families of graphs , 2013, Distributed Computing.

[10]  Leonid Barenboim,et al.  Sublogarithmic distributed MIS algorithm for sparse graphs using Nash-Williams decomposition , 2008, PODC '08.

[11]  Eric Price,et al.  A Hybrid Sampling Scheme for Triangle Counting , 2016, SODA.

[12]  Sergei Vassilvitskii,et al.  Densest Subgraph in Streaming and MapReduce , 2012, Proc. VLDB Endow..

[13]  Lukasz Kowalik,et al.  Adjacency queries in dynamic sparse graphs , 2007, Inf. Process. Lett..

[14]  Seshadhri Comandur,et al.  Linear Time Subgraph Counting, Graph Degeneracy, and the Chasm at Size Six , 2019, ITCS.

[15]  David Eppstein,et al.  Listing All Maximal Cliques in Large Sparse Real-World Graphs , 2011, JEAL.

[16]  Petr A. Golovach,et al.  Parameterized Complexity for Domination Problems on Degenerate Graphs , 2008, WG.

[17]  Dana Ron,et al.  Faster sublinear approximation of the number of k-cliques in low-arboricity graphs , 2020, SODA.

[18]  Frank Fischer,et al.  A Constructive Arboricity Approximation Scheme , 2018, SOFSEM.

[19]  Christos D. Zaroliagis,et al.  Efficient Computation of Implicit Representations of Sparse Graphs , 1997, Discret. Appl. Math..

[20]  Suman Kalyan Bera,et al.  Counting Subgraphs in Degenerate Graphs , 2020, Electron. Colloquium Comput. Complex..

[21]  Dana Ron,et al.  A pr 2 01 6 Sublinear Time Estimation of Degree Distribution Moments : The Arboricity Connection ( Full Version ) Talya Eden , 2016 .

[22]  Clément L. Canonne,et al.  An Adaptivity Hierarchy Theorem for Property Testing , 2017, Electron. Colloquium Comput. Complex..

[23]  Shay Solomon,et al.  Improved Dynamic Graph Coloring , 2018, ESA.

[24]  Sofya Vorotnikova.  Improved 3-pass Algorithm for Counting 4-cycles in Arbitrary Order Streaming , 2020, ArXiv.

[25]  Harold N. Gabow,et al.  Algorithms for graphic polymatroids and parametric s-Sets , 1995, SODA '95.

[26]  Sofya Vorotnikova,et al.  Better Algorithms for Counting Triangles in Data Streams , 2016, PODS.

[27]  Sofya Vorotnikova,et al.  Densest Subgraph in Dynamic Graph Streams , 2015, MFCS.

[28]  W. T. Tutte.  On the Problem of Decomposing a Graph into n Connected Factors , 1961 .

[29]  Sriram V. Pemmaraju,et al.  Distributed graph coloring in a few rounds , 2011, PODC '11.

[30]  Andreas Wiese,et al.  Explicit and Implicit Dynamic Coloring of Graphs with Bounded Arboricity , 2020, ArXiv.

[31]  David Eppstein,et al.  Arboricity and Bipartite Subgraph Listing Algorithms , 1994, Inf. Process. Lett..

[32]  Nikhil Bansal,et al.  Tight approximation bounds for dominating set on graphs of bounded arboricity , 2017, Inf. Process. Lett..

[33]  Moses Charikar,et al.  Greedy approximation algorithms for finding dense components in a graph , 2000, APPROX.

[34]  Norishige Chiba,et al.  Arboricity and Subgraph Listing Algorithms , 1985, SIAM J. Comput..

[35]  C. Nash-Williams.  Edge-disjoint spanning trees of finite graphs , 1961 .

[36]  Charalampos E. Tsourakakis,et al.  Space- and Time-Efficient Algorithm for Maintaining Dense Subgraphs on One-Pass Dynamic Streams , 2015, STOC.

[37]  Raphael Yuster,et al.  Counting Homomorphic Cycles in Degenerate Graphs , 2020, ArXiv.

[38]  Will Rosenbaum,et al.  Almost Optimal Bounds for Sublinear-Time Sampling of k-Cliques: Sampling Cliques is Harder Than Counting , 2020, ArXiv.

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

[40]  Mohsen Ghaffari,et al.  Distributed Arboricity-Dependent Graph Coloring via All-to-All Communication , 2019, ICALP.

[41]  David Peleg,et al.  Local-on-Average Distributed Tasks , 2016, SODA.