Distributed Symmetry Breaking in Hypergraphs
- Shay KuttenAffiliated withFaculty of IE&M, Technion
- , Danupon NanongkaiAffiliated withFaculty of Computer Science, University of Vienna
- , Gopal PanduranganAffiliated withDepartment of Computer Science, University of Houston
- , Peter RobinsonAffiliated withSchool of Computing, National University of Singapore
Fundamental local symmetry breaking problems such as Maximal Independent Set (MIS) and coloring have been recognized as important by the community, and studied extensively in (standard) graphs. In particular, fast (i.e., logarithmic run time) randomized algorithms are well-established for MIS and Δ + 1-coloring in both the LOCAL and CONGEST distributed computing models. On the other hand, comparatively much less is known on the complexity of distributed symmetry breaking in hypergraphs. In particular, a key question is whether a fast (randomized) algorithm for MIS exists for hypergraphs.
In this paper, we study the distributed complexity of symmetry breaking in hypergraphs by presenting distributed randomized algorithms for a variety of fundamental problems under a natural distributed computing model for hypergraphs. We first show that MIS in hypergraphs (of arbitrary dimension) can be solved in O(log2 n) rounds (n is the number of nodes of the hypergraph) in the LOCAL model. We then present a key result of this paper — an O(Δ ε polylog n)-round hypergraph MIS algorithm in the CONGEST model where Δ is the maximum node degree of the hypergraph and ε > 0 is any arbitrarily small constant. We also present distributed algorithms for coloring, maximal matching, and maximal clique in hypergraphs.
To demonstrate the usefulness of hypergraph MIS, we present applications of our hypergraph algorithm to solving problems in (standard) graphs. In particular, the hypergraph MIS yields fast distributed algorithms for the balanced minimal dominating set problem (left open in Harris et al. [ICALP 2013]) and the minimal connected dominating set problem.
Our work shows that while some local symmetry breaking problems such as coloring can be solved in polylogarithmic rounds in both the LOCAL and CONGEST models, for many other hypergraph problems such as MIS, hitting set, and maximal clique, it remains challenging to obtain polylogarithmic time algorithms in the CONGEST model. This work is a step towards understanding this dichotomy in the complexity of hypergraph problems as well as using hypergraphs to design fast distributed algorithms for problems in (standard) graphs.
- Distributed Symmetry Breaking in Hypergraphs
- Book Title
- Distributed Computing
- Book Subtitle
- 28th International Symposium, DISC 2014, Austin, TX, USA, October 12-15, 2014. Proceedings
- pp 469-483
- Print ISBN
- Online ISBN
- Series Title
- Lecture Notes in Computer Science
- Series Volume
- Series ISSN
- Springer Berlin Heidelberg
- Copyright Holder
- Springer-Verlag Berlin Heidelberg
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- Fabian Kuhn (15)
- Editor Affiliations
- 15. Department of Computer Science, University of Freiburg
- Author Affiliations
- 16. Faculty of IE&M, Technion, Haifa, Israel
- 17. Faculty of Computer Science, University of Vienna, Austria
- 18. Department of Computer Science, University of Houston, USA
- 19. School of Computing, National University of Singapore, Singapore
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