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On the Communication Complexity of Finding a King in a Tournament

Authors Nikhil S. Mande , Manaswi Paraashar , Swagato Sanyal, Nitin Saurabh

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Author Details

Nikhil S. Mande
  • University of Liverpool, UK
Manaswi Paraashar
  • University of Copenhagen, Denmark
Swagato Sanyal
  • Indian Institute of Technology Kharagpur, India
Nitin Saurabh
  • Indian Institute of Technology Hyderabad, India

Funding

  • Paraashar, Manaswi: M.P. is supported by ERC grant (QInteract, Grant Agreement No 101078107).
  • Saurabh, Nitin: N.S. was supported by the seed grant (SG/IITH/F285/2022-23/SG-123) from IIT Hyderabad.

Acknowledgements

We thank an anonymous reviewer for pointing out that Theorem 5 implies a better quantum communication upper bound for finding a king than the bound given in an earlier version of our paper.

Cite AsGet BibTex

Nikhil S. Mande, Manaswi Paraashar, Swagato Sanyal, and Nitin Saurabh. On the Communication Complexity of Finding a King in a Tournament. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 64:1-64:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.64

Abstract

A tournament is a complete directed graph. A source in a tournament is a vertex that has no in-neighbours (every other vertex is reachable from it via a path of length 1), and a king in a tournament is a vertex v such that every other vertex is reachable from v via a path of length at most 2. It is well known that every tournament has at least one king. In particular, a maximum out-degree vertex is a king. The tasks of finding a king and a maximum out-degree vertex in a tournament has been relatively well studied in the context of query complexity. We study the communication complexity of finding a king, of finding a maximum out-degree vertex, and of finding a source (if it exists) in a tournament, where the edges are partitioned between two players. The following are our main results for n-vertex tournaments: - We show that the communication task of finding a source in a tournament is equivalent to the well-studied Clique vs. Independent Set (CIS) problem on undirected graphs. As a result, known bounds on the communication complexity of CIS [Yannakakis, JCSS'91, Göös, Pitassi, Watson, SICOMP'18] imply a bound of Θ̃(log² n) for finding a source (if it exists, or outputting that there is no source) in a tournament. - The deterministic and randomized communication complexities of finding a king are Θ(n). The quantum communication complexity of finding a king is Θ̃(√n). - The deterministic, randomized, and quantum communication complexities of finding a maximum out-degree vertex are Θ(n log n), Θ̃(n) and Θ̃(√n), respectively. Our upper bounds above hold for all partitions of edges, and the lower bounds for a specific partition of the edges. One of our lower bounds uses a fooling-set based argument, and all our other lower bounds follow from carefully-constructed reductions from Set-Disjointness. An interesting point to note here is that while the deterministic query complexity of finding a king has been open for over two decades [Shen, Sheng, Wu, SICOMP'03], we are able to essentially resolve the complexity of this problem in a model (communication complexity) that is usually harder to analyze than query complexity.

Subject Classification

ACM Subject Classification
  • Theory of computation → Communication complexity
  • Theory of computation → Quantum complexity theory
  • Theory of computation → Graph algorithms analysis
Keywords
  • Communication complexity
  • tournaments
  • query complexity

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