Linear-in-$$\varDelta $$Δ lower bounds in the LOCAL model

By prior work, there is a distributed graph algorithm that finds a maximal fractional matching (maximal edge packing) in $$O(\varDelta )$$O(Δ) rounds, independently of $$n$$n; here $$\varDelta $$Δ is the maximum degree of the graph and $$n$$n is the number of nodes in the graph. We show that this is optimal: there is no distributed algorithm that finds a maximal fractional matching in $$o(\varDelta )$$o(Δ) rounds, independently of $$n$$n. Our work gives the first linear-in-$$\varDelta $$Δ lower bound for a natural graph problem in the standard $$\mathsf{LOCAL }$$LOCAL model of distributed computing—prior lower bounds for a wide range of graph problems have been at best logarithmic in $$\varDelta $$Δ.

[1]  Thomas Moscibroda,et al.  What Cannot Be Computed Locally , 2004 .

[2]  Roger Wattenhofer,et al.  The price of being near-sighted , 2006, SODA '06.

[3]  Andrzej Czygrinow,et al.  Fast Distributed Approximations in Planar Graphs , 2008, DISC.

[4]  Moni Naor,et al.  What can be computed locally? , 1993, STOC.

[5]  Jukka Suomela,et al.  Deterministic local algorithms, unique identifiers, and fractional graph colouring , 2012, Theor. Comput. Sci..

[6]  Alessandro Panconesi,et al.  On the distributed complexity of computing maximal matchings , 1997, SODA '98.

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

[8]  Mika Göös,et al.  Linear-in-delta lower bounds in the LOCAL model , 2014, PODC '14.

[9]  Michael Luby,et al.  A simple parallel algorithm for the maximal independent set problem , 1985, STOC '85.

[10]  David Peleg,et al.  Distributed Computing: A Locality-Sensitive Approach , 1987 .

[11]  B. H. Neumann,et al.  On Ordered Groups , 1949 .

[12]  Jukka Suomela,et al.  Distributed maximal matching: greedy is optimal , 2012, PODC '12.

[13]  Mika Göös,et al.  No sublogarithmic-time approximation scheme for bipartite vertex cover , 2012, Distributed Computing.

[14]  Christoph Lenzen,et al.  Leveraging Linial's Locality Limit , 2008, DISC.

[15]  Noga Alon,et al.  A Fast and Simple Randomized Parallel Algorithm for the Maximal Independent Set Problem , 1985, J. Algorithms.

[16]  Petteri Kaski,et al.  Local Approximability of Max-Min and Min-Max Linear Programs , 2010, Theory of Computing Systems.

[17]  Roger Wattenhofer,et al.  On the complexity of distributed graph coloring , 2006, PODC '06.

[18]  Leonid Barenboim,et al.  The Locality of Distributed Symmetry Breaking , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[19]  Jukka Suomela,et al.  Lower bounds for local approximation , 2012, PODC '12.

[20]  Dana Angluin,et al.  Local and global properties in networks of processors (Extended Abstract) , 1980, STOC '80.

[21]  Moni Naor,et al.  Local computations on static and dynamic graphs , 1995, Proceedings Third Israel Symposium on the Theory of Computing and Systems.

[22]  Mika Göös,et al.  Linear-in-$Δ$ Lower Bounds in the LOCAL Model , 2013, ArXiv.

[23]  Roger Wattenhofer,et al.  Local Computation , 2010, J. ACM.

[24]  Alon Itai,et al.  A Fast and Simple Randomized Parallel Algorithm for Maximal Matching , 1986, Inf. Process. Lett..

[25]  Alessandro Panconesi,et al.  Some simple distributed algorithms for sparse networks , 2001, Distributed Computing.

[26]  Valentin Polishchuk,et al.  A Local 2-Approximation Algorithm for the Vertex Cover Problem , 2009, DISC.

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

[28]  Jukka Suomela,et al.  Fast distributed approximation algorithms for vertex cover and set cover in anonymous networks , 2010, SPAA '10.