Abstract
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.
- Yehuda Afek, Shay Kutten, and Moti Yung. 1990. Memory-efficient self stabilizing protocols for general networks. In WDAG (Lecture Notes in Computer Science), Jan van Leeuwen and Nicola Santoro (Eds.), Vol. 486. Springer, Berlin, 15--28.Google Scholar
- J. Akiyama, H. Era, and F. Harary. 1983. Regular graphs containing a given graph. Elem. Math. 83 83 (1983), 15--17.Google Scholar
- Noga Alon, Laszlo Babai, and Alon Itai. 1986. A fast and simple randomized parallel algorithm for the maximal independent set problem. J. Algorithm. 7, 4 (1986), 567--583.Google ScholarDigital Library
- D. Angluin and A. Gardiner. 1981. Finite common coverings of pairs of regular graphs. J. Comb. Theor. Ser. B 30, 2 (1981), 184--187.Google ScholarCross Ref
- Baruch Awerbuch and Michael Sipser. 1988. Dynamic networks are as fast as static networks. In Proc. Symp. Foundations of Computer Science (FOCS). 206--219.Google ScholarDigital Library
- Baruch Awerbuch and George Varghese. 1991. Distributed program checking: A paradigm for building self-stabilizing distributed protocols. In Proc. Symp. on Foundations of Computer Science (FOCS). 258--267.Google ScholarDigital Library
- R. Bar-Yehuda, K. Censor-Hillel, and G. Schwartzman. 2016. A distributed (2 + ε)-approximation for vertex cover in O(log Δ/εlog log Δ) rounds. CoRR abs/1602.03713v2.Google Scholar
- L. Barenboim and M. Elkin. 2013. Distributed Graph Coloring: Fundamentals and Recent Developments. Morgan & Claypool Publishers.Google Scholar
- Yair Bartal, John W. Byers, and Danny Raz. 1997. Global optimization using local information with applications to flow control. In Proc. Symp. on Foundations of Computer Science (FOCS). 303--312.Google ScholarCross Ref
- B. Bollobas. 1978. Extremal Graph Theory. Academic Press.Google Scholar
- R. Cole and U. Vishkin. 1986. Deterministic coin tossing with applications to optimal parallel list ranking. Inform. Control 70, 1 (1986), 32--53.Google ScholarDigital Library
- Andrzej Czygrinow, Michał Hańćkowiak, and Wojciech Wawrzyniak. 2008. Fast distributed approximations in planar graphs. In Proc. 22nd Symp. on Distributed Computing (DISC). 78--92.Google ScholarDigital Library
- Erik D. Demaine, Uriel Feige, Mohammad Taghi Hajiaghayi, and Mohammad R. Salavatipour. 2006. Combination can be hard: Approximability of the unique coverage problem. In Proc. of the 17th ACM-SIAM Symposium on Discrete Algorithm (SODA). 162--171.Google Scholar
- Edsger W. Dijkstra. 1974. Self-stabilizing systems in spite of distributed control. Communications of the ACM 17, 11 (1974), 643--644. DOI:http://dx.doi.org/10.1145/361179.361202Google ScholarDigital Library
- D. Dubhashi, A. Mei, A. Panconesi, J. Radhakrishnan, and A. Srinivasan. 2003. Fast distributed algorithms for (weakly) connected dominating sets and linear-size skeletons. In Proc. of the ACM-SIAM Symposium on Discrete Algorithms (SODA). 717--724.Google Scholar
- Michael Elkin. 2004a. Distributed approximation—A survey. ACM SIGACT News—Distributed Computing Column 35, 4 (2004).Google Scholar
- Michael Elkin. 2004b. An unconditional lower bound on the hardness of approximation of distributed minimum spanning tree problem. In Proc. of the 36th ACM Symposium on Theory of Computing (STOC). 331--340.Google Scholar
- P. Erdős and H. Sachs. 1963. Reguläre graphen gegebener taillenweite mit minimaler knotenzahl. Wiss. Z. Martin-Luther-U. Halle Math.-Nat. 12 (1963), 251--257.Google Scholar
- Uriel Feige, Magnús M. Halldórsson, Guy Kortsarz, and Aravind Srinivasan. 2003. Approximating the domatic number. SIAM J. Comput. 32, 1 (2003), 172--195.Google ScholarDigital Library
- F. Fich and E. Ruppert. 2003. Hundreds of impossibility results for distributed computing. Distrib. Comput. 16, 2--3 (2003), 121--163.Google ScholarDigital Library
- Michael J. Fischer, Nancy A. Lynch, and Michael S. Paterson. 1985. Impossibility of distributed consensus with one faulty process. J. ACM 32, 2 (1985), 374--382.Google ScholarDigital Library
- L. Fleischer. 2000. Approximating fractional multicommodity flow independent of the number of commodities. SIAM J. Discr. Math. 13, 4 (2000), 505--520.Google ScholarDigital Library
- P. Fraigniaud, A. Korman, and D. Peleg. 2013. Towards a complexity theory for local distributed computing. J. ACM 60, 5 (2013), 35.Google ScholarDigital Library
- Seth Copen Goldstein, Jason D. Campbell, and Todd C. Mowry. 2005. Programmable matter. Computer 38, 6 (2005), 99--101. DOI:http://dx.doi.org/10.1109/MC.2005.198Google ScholarDigital Library
- A. Israeli and A. Itai. 1986. A fast and simple randomized parallel algorithm for maximal matching. Inform. Process. Lett. 22 (1986), 77--80.Google ScholarCross Ref
- L. Jia, R. Rajaraman, and R. Suel. 2001. An efficient distributed algorithm for constructing small dominating sets. In Proc. of the 20th ACM Symposium on Principles of Distributed Computing (PODC). 33--42.Google Scholar
- Fabian Kuhn, Thomas Moscibroda, and Roger Wattenhofer. 2004. What cannot be computed locally!. In Proc. of the 23rd ACM Symposium on the Principles of Distributed Computing (PODC). 300--309.Google Scholar
- Fabian Kuhn, Thomas Moscibroda, and Roger Wattenhofer. 2006. The price of being near-sighted. In Proc. of the 17th ACM-SIAM Symposium on Discrete Algorithms (SODA).Google ScholarCross Ref
- F. Kuhn and R. Wattenhofer. 2003. Constant-time distributed dominating set approximation. In Proc. of the 22nd Annual ACM Symp. on Principles of Distributed Computing (PODC). 25--32.Google Scholar
- Leslie Lamport, Robert Shostak, and Marshall Pease. 1982. The byzantine generals problem. ACM Trans. Program. Lang. Syst. 4, 3 (1982), 382--401.Google ScholarDigital Library
- F. Lazebnik and V. A. Ustimenko. 1995. Explicit construction of graphs with an arbitrary large girth and of large size. Discr. Appl. Math. 60, 1--3 (1995), 275--284.Google ScholarDigital Library
- Christoph Lenzen, Yvonne Anne Oswald, and Roger Wattenhofer. 2008. What can be approximated locally? In 20th ACM Symposium on Parallelism in Algorithms and Architecture (SPAA), Munich, Germany.Google ScholarDigital Library
- Christoph Lenzen, Jukka Suomela, and Roger Wattenhofer. 2009. Local algorithms: Self-stabilization on speed. In 11th International Symposium on Stabilization, Safety, and Security of Distributed Systems (SSS). 17--34.Google ScholarDigital Library
- Christoph Lenzen and Roger Wattenhofer. 2008. Leveraging linial’s locality limit. In Proc. 22nd Symp. on Distributed Computing (DISC). 394--407.Google ScholarDigital Library
- Nathan Linial. 1992. Locality in distributed graph algorithms. SIAM J. Comput. 21, 1 (1992), 193--201.Google ScholarDigital Library
- N. Linial and M. Saks. 1993. Low diameter graph decompositions. Combinatorica 13, 4 (1993), 441--454.Google ScholarCross Ref
- M. Luby. 1986. A simple parallel algorithm for the maximal independent set problem. SIAM J. Comput. 15 (1986), 1036--1053.Google ScholarDigital Library
- Grzegorz Malewicz, Matthew H. Austern, Aart J. C. Bik, James C. Dehnert, Ilan Horn, Naty Leiser, and Grzegorz Czajkowski. 2010. Pregel: A system for large-scale graph processing. In Proceedings of the International Conference on Management of Data (SIGMOD).Google ScholarDigital Library
- Moni Naor and Larry Stockmeyer. 1995. What can be computed locally? SIAM J. Comput. 24, 6 (1995), 1259--1277.Google ScholarDigital Library
- Huy N. Nguyen and Krzysztof Onak. 2008. Constant-time approximation algorithms via local improvements. In Proc. of the 49th Symposium on Foundations of Computer Science (FOCS). 327--336.Google Scholar
- C. H. Papadimitriou and M. Yannakakis. 1991. On the value of information in distributed decision making. In Proc. of the 10th ACM Symposium on Principles of Distributed Computing (PODC). 61--64.Google Scholar
- C. H. Papadimitriou and M. Yannakakis. 1993. Linear programming without the matrix. In Proc. of the 25th ACM Symposium on Theory of Computing (STOC). 121--129.Google Scholar
- Michal Parnas and Dana Ron. 2007. Approximating the minimum vertex cover in sublinear time and a connection to distributed algorithms. Theor. Comput. Sci. 381, 1-3 (2007), 183--196.Google ScholarDigital Library
- David Peleg. 2000. Distributed Computing: A Locality-Sensitive Approach. SIAM Monographs on Discrete Mathematics and Applications. SIAM, Philadelpia, PA.Google ScholarDigital Library
- S. Plotkin, D. Shmoys, and E. Tardos. 1995. Fast approximation algorithms for fractional packing and covering problems. Math. Operat. Res. 20 (1995), 257--301.Google ScholarDigital Library
- S. Rajagopalan and V. Vazirani. 1998. Primal-dual RNC approximation algorithms for set cover and covering integer programs. SIAM J. Comput. 28 (1998), 525--540.Google ScholarDigital Library
- Atish Das Sarma, Stephan Holzer, Liah Kor, Amos Korman, Danupon Nanongkai, Gopal Pandurangan, David Peleg, and Roger Wattenhofer. 2012. Distributed verification and hardness of distributed approximation. SIAM J. Computing 41, 5 (2012), 1235--1265.Google ScholarCross Ref
- Johannes Schneider and Roger Wattenhofer. 2008. A log-star distributed maximal independent set algorithm for growth-bounded graphs. In 27th ACM Symposium on Principles of Distributed Computing (PODC), Toronto, Canada. ACM, New York, NY, 35--44.Google ScholarDigital Library
- Johannes Schneider and Roger Wattenhofer. 2011. Bounds on contention management algorithms. Theoretical Computer Science 412, 32 (2011), 4151--4160.Google ScholarDigital Library
- A. Srinivasan. 1995. Improved approximations of packing and covering problems. In Proc. of the 27th ACM Symposium on Theory of Computing (STOC). ACM, New York, NY, 268--276.Google ScholarDigital Library
- Aaron Sterling. 2009. Memory consistency conditions for self-assembly programming. In 1st Innovations in Computer Science (ICS). Beijing, China, 490--500.Google Scholar
- J. Suomela. 2011. Survey of local algorithms. ACM Computing Surveys 45, 2 (2011), 24:1--24:40.Google Scholar
- Mirjam Wattenhofer and Roger Wattenhofer. 2004. Distributed weighted matching. In Proc. of the 18th Conference on Distributed Computing (DISC). 335--348.Google ScholarCross Ref
- N. E. Young. 2001. Sequential and parallel algorithms for mixed packing and covering. In Proc. of the 42nd Symposium on Foundations of Computer Science (FOCS). 538--546.Google ScholarCross Ref
Index Terms
- Local Computation: Lower and Upper Bounds
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