Local Computation

The question of what can be computed, and how efficiently, is at the core of computer science. Not surprisingly, in distributed systems and networking research, an equally fundamental question is what can be computed in a distributed fashion. More precisely, if nodes of a network must base their decision on information in their local neighborhood only, how well can they compute or approximate a global (optimization) problem? In this paper we give the first polylogarithmic lower bound on such local computation for (optimization) problems including minimum vertex cover, minimum (connected) dominating set, maximum matching, maximal independent set, and maximal matching. In addition, we present a new distributed algorithm for solving general covering and packing linear programs. For some problems this algorithm is tight with the lower bounds, whereas for others it is a distributed approximation scheme. Together, our lower and upper bounds establish the local computability and approximability of a large class of problems, characterizing how much local information is required to solve these tasks.

[1]  Vijay V. Vazirani,et al.  Primal-Dual RNC Approximation Algorithms for Set Cover and Covering Integer Programs , 1999, SIAM J. Comput..

[2]  Aravind Srinivasan,et al.  Improved approximations of packing and covering problems , 1995, STOC '95.

[3]  Roger Wattenhofer,et al.  Bounds on Contention Management Algorithms , 2009, ISAAC.

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

[5]  B. Awerbuch,et al.  Distributed program checking: a paradigm for building self-stabilizing distributed protocols , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[6]  Edsger W. Dijkstra,et al.  Self stabilization in spite of distributed control , 1974 .

[7]  Danny Raz,et al.  Global optimization using local information with applications to flow control , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

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

[9]  Leslie Lamport,et al.  The Byzantine Generals Problem , 1982, TOPL.

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

[11]  Michael Elkin,et al.  Distributed approximation: a survey , 2004, SIGA.

[12]  Neal E. Young,et al.  Sequential and parallel algorithms for mixed packing and covering , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[13]  Christoph Lenzen,et al.  Local Algorithms: Self-stabilization on Speed , 2009, SSS.

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

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

[16]  C. Maheswari,et al.  Some (r, 2, K)-regular Graphs Containing A Given Graph. , 2012 .

[17]  Felix Lazebnik,et al.  Explicit Construction of Graphs with an Arbitrary Large Girth and of Large Size , 1995, Discret. Appl. Math..

[18]  Aart J. C. Bik,et al.  Pregel: a system for large-scale graph processing , 2010, SIGMOD Conference.

[19]  Jukka Suomela,et al.  Survey of local algorithms , 2013, CSUR.

[20]  Vijay V. Vazirani,et al.  Approximation Algorithms , 2001, Springer Berlin Heidelberg.

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

[22]  Reuven Bar-Yehuda,et al.  A Distributed (2+ε)-Approximation for Vertex Cover in O(logδ/ε log log δ) Rounds , 2016, PODC.

[23]  Mihalis Yannakakis,et al.  Linear programming without the matrix , 1993, STOC.

[24]  Reuven Bar-Yehuda,et al.  Fe b 20 16 A Distributed ( 2 + ǫ )-Approximation for Vertex Cover in O ( log ∆ ǫ log log ∆ ) , 2016 .

[25]  Mihalis Yannakakis,et al.  On the value of information in distributed decision-making (extended abstract) , 1991, PODC '91.

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

[27]  Edsger W. Dijkstra,et al.  Self-stabilizing systems in spite of distributed control , 1974, CACM.

[28]  W. McCarthy.  Programmable matter , 2000, Nature.

[29]  Dana Ron,et al.  On Approximating the Minimum Vertex Cover in Sublinear Time and the Connection to Distributed Algorithms , 2007, Electron. Colloquium Comput. Complex..

[30]  H. Sachs,et al.  Regukre Graphen gegebener Taillenweite mit minimaler Knotenzahl , 1963 .

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

[32]  Faith Ellen,et al.  Hundreds of impossibility results for distributed computing , 2003, Distributed Computing.

[33]  Baruch Awerbuch,et al.  Dynamic networks are as fast as static networks , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

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

[35]  Roger Wattenhofer,et al.  A log-star distributed maximal independent set algorithm for growth-bounded graphs , 2008, PODC '08.

[36]  F. Lazebnik,et al.  A new series of dense graphs of high girth , 1995, math/9501231.

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

[38]  Roger Wattenhofer,et al.  Distributed Weighted Matching , 2004, DISC.

[39]  Christoph Lenzen,et al.  What can be approximated locally?: case study: dominating sets in planar graphs , 2008, SPAA '08.

[40]  Moti Yung,et al.  Memory-Efficient Self Stabilizing Protocols for General Networks , 1990, WDAG.

[41]  Roger Wattenhofer,et al.  Constant-time distributed dominating set approximation , 2003, PODC '03.

[42]  Lujun Jia,et al.  An efficient distributed algorithm for constructing small dominating sets , 2002, Distributed Computing.

[43]  Uriel Feige,et al.  Approximating the domatic number , 2000, STOC '00.

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

[45]  Dana Angluin,et al.  Finite common coverings of pairs of regular graphs , 1981, J. Comb. Theory B.

[46]  Erik D. Demaine,et al.  Combination can be hard: approximability of the unique coverage problem , 2006, SODA '06.

[47]  Krzysztof Onak,et al.  Constant-Time Approximation Algorithms via Local Improvements , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[48]  A. Shapira,et al.  Extremal Graph Theory , 2013 .

[49]  Nancy A. Lynch,et al.  Impossibility of distributed consensus with one faulty process , 1983, PODS '83.

[50]  Roger Wattenhofer,et al.  What cannot be computed locally! , 2004, PODC '04.

[51]  Éva Tardos,et al.  Fast approximation algorithms for fractional packing and covering problems , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[52]  Dan Suciu,et al.  Journal of the ACM , 2006 .

[53]  M. Kaufmann.  What Can Be Computed Locally ? , 2003 .

[54]  Aaron Sterling,et al.  Memory Consistency Conditions for Self-Assembly Programming , 2009, ICS.

[55]  Pierre Fraigniaud,et al.  Towards a complexity theory for local distributed computing , 2013, JACM.

[56]  Aravind Srinivasan,et al.  Fast distributed algorithms for (weakly) connected dominating sets and linear-size skeletons , 2003, J. Comput. Syst. Sci..

[57]  Lisa Fleischer,et al.  Approximating Fractional Multicommodity Flow Independent of the Number of Commodities , 2000, SIAM J. Discret. Math..

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