On the Locality of Nash-Williams Forest Decomposition and Star-Forest Decomposition

Given a graph $G=(V,E)$ with arboricity $\alpha$, we study the problem of decomposing the edges of $G$ into $(1+\epsilon)\alpha$ disjoint forests in the distributed LOCAL model. Barenboim and Elkin [PODC `08] gave a LOCAL algorithm that computes a $(2+\epsilon)\alpha$-forest decomposition using $O(\frac{\log n}{\epsilon})$ rounds. Ghaffari and Su [SODA `17] made further progress by computing a $(1+\epsilon) \alpha$-forest decomposition in $O(\frac{\log^3 n}{\epsilon^4})$ rounds when $\epsilon \alpha = \Omega(\sqrt{\alpha \log n})$, i.e. the limit of their algorithm is an $(\alpha+ \Omega(\sqrt{\alpha \log n}))$-forest decomposition. This algorithm, based on a combinatorial construction of Alon, McDiarmid \& Reed [Combinatorica `92], in fact provides a decomposition of the graph into \emph{star-forests}, i.e. each forest is a collection of stars. Our main result in this paper is to reduce the threshold of $\epsilon \alpha$ in $(1+\epsilon)\alpha$-forest decomposition and star-forest decomposition. This further answers the $10^{\text{th}}$ open question from Barenboim and Elkin's {\it Distributed Graph Algorithms} book. Moreover, it gives the first $(1+\epsilon)\alpha$-orientation algorithms with {\it linear dependencies} on $\epsilon^{-1}$. At a high level, our results for forest-decomposition are based on a combination of network decomposition, load balancing, and a new structural result on local augmenting sequences. Our result for star-forest decomposition uses a more careful probabilistic analysis for the construction of Alon, McDiarmid, \& Reed; the bounds on star-arboricity here were not previously known, even non-constructively.

[1]  Silvio Lattanzi,et al.  Improved Parallel Algorithms for Density-Based Network Clustering , 2019, ICML.

[2]  Václav Rozhon,et al.  Polylogarithmic-time deterministic network decomposition and distributed derandomization , 2019, STOC.

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

[4]  Richard M. Karp,et al.  Massively Parallel Computation of Matching and MIS in Sparse Graphs , 2019, PODC.

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

[6]  Richard Cole,et al.  Deterministic Coin Tossing with Applications to Optimal Parallel List Ranking , 2018, Inf. Control..

[7]  Kamesh Munagala,et al.  Efficient Primal-Dual Graph Algorithms for MapReduce , 2014, WAW.

[8]  David G. Harris Distributed Local Approximation Algorithms for Maximum Matching in Graphs and Hypergraphs , 2019, 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS).

[9]  Alessandro Panconesi,et al.  Near-Optimal, Distributed Edge Colouring via the Nibble Method , 1996, Theor. Comput. Sci..

[10]  Hsin-Hao Su,et al.  Distributed algorithms for the Lovász local lemma and graph coloring , 2014, Distributed Computing.

[11]  Gerth Stølting Brodal,et al.  A Simple Greedy Algorithm for Dynamic Graph Orientation , 2018, Algorithmica.

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

[13]  Nathan Linial,et al.  Locality in Distributed Graph Algorithms , 1992, SIAM J. Comput..

[14]  Yasukazu Aoki The star-arboricity of the complete regular multipartite graphs , 1990, Discret. Math..

[15]  Harold N. Gabow,et al.  Forests, frames, and games: algorithms for matroid sums and applications , 1988, STOC '88.

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

[17]  Leonid Barenboim,et al.  Deterministic Distributed Vertex Coloring in Polylogarithmic Time , 2010, JACM.

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

[19]  Gary L. Miller,et al.  Parallel graph decompositions using random shifts , 2013, SPAA.

[20]  Fabian Kuhn,et al.  On the complexity of local distributed graph problems , 2016, STOC.

[21]  N Linial,et al.  Low diameter graph decompositions , 1993, Comb..

[22]  Hsin-Hao Su,et al.  Distributed Dense Subgraph Detection and Low Outdegree Orientation , 2019, DISC.

[23]  Hsin-Hao Su,et al.  (2Δ - l)-Edge-Coloring is Much Easier than Maximal Matching in the Distributed Setting , 2015, SODA.

[24]  Ashwin Lall,et al.  Dense Subgraphs on Dynamic Networks , 2012, DISC.

[25]  Hsin-Hao Su,et al.  Distributed Degree Splitting, Edge Coloring, and Orientations , 2016, SODA.

[26]  Lenore Cowen,et al.  Fast Distributed Network Decompositions and Covers , 1996, J. Parallel Distributed Comput..

[27]  Fabian Kuhn Faster Deterministic Distributed Coloring Through Recursive List Coloring , 2020, SODA.

[28]  Subramanian Ramanathan,et al.  Scheduling algorithms for multihop radio networks , 1993, TNET.

[29]  Fabian Kuhn,et al.  Deterministic Distributed Edge-Coloring via Hypergraph Maximal Matching , 2017, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).

[30]  Aravind Srinivasan,et al.  Randomized Distributed Edge Coloring via an Extension of the Chernoff-Hoeffding Bounds , 1997, SIAM J. Comput..

[31]  Gerth Stølting Brodal,et al.  Dynamic Representation of Sparse Graphs , 1999, WADS.

[32]  Julian Shun,et al.  Parallel Clique Counting and Peeling Algorithms , 2020, ACDA.

[33]  Matthias F. Stallmann,et al.  Efficient Algorithms for Graphic Matroid Intersection and Parity (Extended Abstract) , 1985, ICALP.

[34]  Anton Bernshteyn,et al.  A Fast Distributed Algorithm for (Δ+1)-Edge-Coloring , 2020, J. Comb. Theory, Ser. B.

[35]  Leonid Barenboim,et al.  Distributed Graph Coloring: Fundamentals and Recent Developments , 2013, Distributed Graph Coloring: Fundamentals and Recent Developments.

[36]  Fabian Kuhn,et al.  Deterministic distributed edge-coloring with fewer colors , 2017, STOC.

[37]  Paul D. Seymour A Note on List Arboricity , 1998, J. Comb. Theory, Ser. B.

[38]  Michael Elkin,et al.  Distributed Strong Diameter Network Decomposition: Extended Abstract , 2016, PODC.

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

[40]  Andrew V. Goldberg,et al.  Finding a Maximum Density Subgraph , 1984 .

[41]  Fabian Kuhn,et al.  On Derandomizing Local Distributed Algorithms , 2017, 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS).

[42]  Robert E. Tarjan,et al.  A Note on Finding Minimum-Cost Edge-Disjoint Spanning Trees , 1985, Math. Oper. Res..

[43]  Lukasz Kowalik,et al.  Approximation Scheme for Lowest Outdegree Orientation and Graph Density Measures , 2006, ISAAC.

[44]  Samir Khuller,et al.  On Finding Dense Subgraphs , 2009, ICALP.

[45]  Saurabh Sawlani,et al.  Near-optimal fully dynamic densest subgraph , 2019, STOC.

[46]  Robert E. Tarjan,et al.  A Fast Parametric Maximum Flow Algorithm and Applications , 1989, SIAM J. Comput..

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

[48]  Seth Pettie,et al.  The Complexity of Distributed Edge Coloring with Small Palettes , 2017, SODA.

[49]  H. Imai NETWORK-FLOW ALGORITHMS FOR LOWER-TRUNCATED TRANSVERSAL POLYMATROIDS , 1983 .

[50]  Amit Kumar,et al.  Maintaining Assignments Online: Matching, Scheduling, and Flows , 2014, SODA.

[51]  Abhay Karandikar,et al.  On High Spatial Reuse Link Scheduling in STDMA Wireless Ad Hoc Networks , 2007, IEEE GLOBECOM 2007 - IEEE Global Telecommunications Conference.

[52]  Hsin-Hao Su,et al.  Towards the locality of Vizing’s theorem , 2019, STOC.

[53]  Noga Alon,et al.  The star arboricity of graphs , 1988, Discret. Math..