Abstract
The Minimum Dominating Set (MDS) problem is a fundamental and challenging problem in distributed computing. While it is well known that minimum dominating sets cannot be well approximated locally on general graphs, in recent years there has been much progress on computing good local approximations on sparse graphs and in particular on planar graphs. In this article, we study distributed and deterministic MDS approximation algorithms for graph classes beyond planar graphs. In particular, we show that existing approximation bounds for planar graphs can be lifted to bounded genus graphs and more general graphs, which we call locally embeddable graphs, and present
(1) a local constant-time, constant-factor MDS approximation algorithm on locally embeddable graphs, and
(2) a local O(log*n)-time (1+ϵ)-approximation scheme for any ϵ > 0 on graphs of bounded genus.
Our main technical contribution is a new analysis of a slightly modified variant of an existing algorithm by Lenzen et al. [21]. Interestingly, unlike existing proofs for planar graphs, our analysis does not rely on direct topological arguments but on combinatorial density arguments only.
- Saeed Akhoondian Amiri, Patrice Ossona de Mendez, Roman Rabinovich, and Sebastian Siebertz. 2018. Distributed domination on graph classes of bounded expansion. In Proceedings of the 30th Symposium on Parallelism in Algorithms and Architectures. ACM, 143--151.Google Scholar
- Saeed Akhoondian Amiri. 2017. Structural Graph Theory Meets Algorithms: Covering and Connectivity Problems in Graphs. Ph.D. Dissertation. Logic and Semantic Group, Technical University Berlin.Google Scholar
- Saeed Akhoondian Amiri and Stefan Schmid. 2016. Brief announcement: A log-time local MDS approximation scheme for bounded genus graphs. In Proceedings of the 30th International Symposium on Distributed Computing (DISC’16). Springer.Google Scholar
- Saeed Akhoondian Amiri, Stefan Schmid, and Sebastian Siebertz. 2016. A local constant factor MDS approximation for bounded genus graphs. In Proceedings of the ACM Symposium on Principles of Distributed Computing (PODC’16).Google Scholar
- Brenda S. Baker. 1994. Approximation algorithms for NP-complete problems on planar graphs. J. ACM 41, 1 (Jan. 1994), 153--180. Google ScholarDigital Library
- Nikhil Bansal and Seeun William Umboh. 2017. Tight approximation bounds for dominating set on graphs of bounded arboricity. Info. Process. Lett. 122 (2017), 21--24.Google ScholarCross Ref
- Leonid Barenboim, Michael Elkin, and Cyril Gavoille. 2014. A fast network-decomposition algorithm and its applications to constant-time distributed computation. In Proceedings of the International Colloquium on Structural Information and Communication Complexity (SIROCCO’14). 209--223.Google Scholar
- Hervé Brönnimann and Michael T. Goodrich. 1995. Almost optimal set covers in finite VC-dimension. Discrete Comput. Geom. 14, 4 (1995), 463--479. Google ScholarDigital Library
- Andrzej Czygrinow, Michal Hańćkowiak, and Wojciech Wawrzyniak. 2008. Fast distributed approximations in planar graphs. In Proceedings of the 22nd International Symposium on Distributed Computing (DISC’08). 78--92.Google ScholarDigital Library
- Irit Dinur and David Steurer. 2014. Analytical approach to parallel repetition. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing. ACM, 624--633. Google ScholarDigital Library
- Yuval Emek and Roger Wattenhofer. 2013. Stone age distributed computing. In Proceedings of the ACM Symposium on Principles of Distributed Computing (PODC’13). 137--146.Google ScholarDigital Library
- David Eppstein. 2000. Diameter and treewidth in minor-closed graph families. Algorithmica 27, 3--4 (2000), 275--291.Google ScholarCross Ref
- Guy Even, Dror Rawitz, and Shimon Moni Shahar. 2005. Hitting sets when the VC-dimension is small. Info. Process. Lett. 95, 2 (2005), 358--362. Google ScholarDigital Library
- Michael R. Garey and David S. Johnson. 1979. Computers and Intractability: A Guide to the Theory of NP-completeness. W. H. Freeman, San Francisco. Google ScholarDigital Library
- Martin Grohe. 2003. Local tree-width, excluded minors, and approximation algorithms. Combinatorica 23, 4 (2003), 613--632. Google ScholarDigital Library
- Sariel Har-Peled and Kent Quanrud. 2015. Approximation algorithms for polynomial-expansion and low-density graphs. In Proceedings of the European Symposium on Algorithms (ESA’15). Springer, 717--728.Google ScholarCross Ref
- Miikka Hilke, Christoph Lenzen, and Jukka Suomela. 2014. Brief announcement: Local approximability of minimum dominating set on planar graphs. In Proceedings of the ACM Symposium on Principles of Distributed Computing (PODC’14). 344--346.Google ScholarDigital Library
- David S. Johnson. 1974. Approximation algorithms for combinatorial problems. J. Comput. Syst. Sci. 9, 3 (1974), 256--278. Google ScholarDigital Library
- Richard M. Karp. 1972. Reducibility among combinatorial problems. In Complexity of Computer Computations. Springer, 85--103.Google Scholar
- Fabian Kuhn, Thomas Moscibroda, and Roger Wattenhofer. 2016. Local computation: Lower and upper bounds. J. ACM 63, 2, Article 17 (Mar. 2016). Google ScholarDigital Library
- Christoph Lenzen, Yvonne Anne Pignolet, and Roger Wattenhofer. 2013. Distributed minimum dominating set approximations in restricted families of graphs. Distrib. Comput. 26, 2 (2013), 119--137.Google ScholarDigital Library
- Christoph Lenzen and Roger Wattenhofer. 2008. Leveraging Linial’s locality limit. In Distributed Computing. Lecture Notes in Computer Science, Vol. 5218. Springer, 394--407.Google Scholar
- Christoph Lenzen and Roger Wattenhofer. 2010. Minimum dominating set approximation in graphs of bounded arboricity. In Distributed Computing, vol. 6343. Springer, 510--524.Google Scholar
- Nathan Linial. 1992. Locality in distributed graph algorithms. SIAM J. Comput. 21, 1 (Feb. 1992), 193--201. Google ScholarDigital Library
- László Lovász. 1975. On the ratio of optimal integral and fractional covers. Discrete Math. 13, 4 (1975), 383--390. Google ScholarDigital Library
- Bojan Mohar and Carsten Thomassen. 2001. Graphs on Surfaces. Johns Hopkins University Press.Google Scholar
- Jaroslav Nešetřil and Patrice Ossona De Mendez. 2012. Sparsity: Graphs, Structures, and Algorithms, Vol. 28. Springer Science 8 Business Media. Google ScholarDigital Library
- Sebastian Siebertz. 2019. Greedy domination on biclique-free graphs. Info. Process. Lett. 145 (2019), 64--67.Google ScholarDigital Library
- Jukka Suomela. 2013. Survey of local algorithms. ACM Comput. Surv. 45, 2, Article 24 (Mar. 2013), 24:1--24:40. Google ScholarDigital Library
- Wojciech Wawrzyniak. 2013. Brief announcement: A local approximation algorithm for MDS problem in anonymous planar networks. In Proceedings of the ACM Symposium on Principles of Distributed Computing (PODC’13). ACM, 406--408.Google ScholarDigital Library
- Wojciech Wawrzyniak. 2014. A strengthened analysis of a local algorithm for the minimum dominating set problem in planar graphs. Proc. Info. Process. Lett. 114, 3 (2014), 94--98.Google ScholarDigital Library
Index Terms
- Distributed Dominating Set Approximations beyond Planar Graphs
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