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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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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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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.

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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

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.

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