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On the communication complexity of finding a king in a tournament

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

TL;DR

The paper resolves the two-party communication complexity of three central tournament problems by linking source-finding to the CIS problem, and by establishing tight bounds for KING_n and MOD_n across deterministic, randomized, and quantum models. It achieves this via reductions from Set-Disjointness, the PMF/IndexKING problems, and the BCW quantum-classical simulation framework, plus a novel class of tournaments G_{S,\sigma} to drive lower bounds. The results show D^{cc}(SRC_E) = \tilde{\Theta}(\log^2 n), D^{cc}(KING_n) = R^{cc}(KING_n) = \Theta(n) with Q^{cc}(KING_n) = \tilde{\Theta}(\sqrt{n}), and D^{cc}(MOD_n) = \Theta(n \log n), R^{cc}(MOD_n) = \tilde{Theta}(n), Q^{cc}(MOD_n) = \tilde{Theta}(\sqrt{n}). These findings illuminate fundamental separations between deterministic, randomized, and quantum communication in tournament settings, and connect natural graph problems to established CIS and Set-Disjointness frameworks with implications for related decision-tree measures.

Abstract

A tournament is a complete directed graph. 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, one of which is a maximum out-degree vertex. The tasks of finding a king, a maximum out-degree vertex and a source in a tournament has been relatively well studied in the context of query complexity. We study the communication complexity of these tasks, where the edges are partitioned between two players. The following are our main results for n-vertex tournaments: 1) The deterministic communication complexity of finding whether a source exists is tilde{Theta}(log^2 n). 2) The deterministic and randomized communication complexities of finding a king are Theta(n). The quantum communication complexity is tilde{Theta}(sqrt{n}). 3) The deterministic, randomized and quantum communication complexities of finding a maximum out-degree vertex are Theta(n log n), tilde{Theta}(n) and tilde{Theta}(sqrt{n}), respectively. Our upper bounds hold for all partitions of edges, and the lower bounds for a specific partition of the edges. To show the first bullet above, we show, perhaps surprisingly, that finding a source in a tournament is equivalent to the well-studied Clique vs. Independent Set (CIS) problem on undirected graphs. Our bounds for finding a source then follow from known bounds on the complexity of the CIS problem. In view of this equivalence, we can view the task of finding a king in a tournament to be a natural generalization of CIS. 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.

On the communication complexity of finding a king in a tournament

TL;DR

The paper resolves the two-party communication complexity of three central tournament problems by linking source-finding to the CIS problem, and by establishing tight bounds for KING_n and MOD_n across deterministic, randomized, and quantum models. It achieves this via reductions from Set-Disjointness, the PMF/IndexKING problems, and the BCW quantum-classical simulation framework, plus a novel class of tournaments G_{S,\sigma} to drive lower bounds. The results show D^{cc}(SRC_E) = \tilde{\Theta}(\log^2 n), D^{cc}(KING_n) = R^{cc}(KING_n) = \Theta(n) with Q^{cc}(KING_n) = \tilde{\Theta}(\sqrt{n}), and D^{cc}(MOD_n) = \Theta(n \log n), R^{cc}(MOD_n) = \tilde{Theta}(n), Q^{cc}(MOD_n) = \tilde{Theta}(\sqrt{n}). These findings illuminate fundamental separations between deterministic, randomized, and quantum communication in tournament settings, and connect natural graph problems to established CIS and Set-Disjointness frameworks with implications for related decision-tree measures.

Abstract

A tournament is a complete directed graph. 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, one of which is a maximum out-degree vertex. The tasks of finding a king, a maximum out-degree vertex and a source in a tournament has been relatively well studied in the context of query complexity. We study the communication complexity of these tasks, where the edges are partitioned between two players. The following are our main results for n-vertex tournaments: 1) The deterministic communication complexity of finding whether a source exists is tilde{Theta}(log^2 n). 2) The deterministic and randomized communication complexities of finding a king are Theta(n). The quantum communication complexity is tilde{Theta}(sqrt{n}). 3) The deterministic, randomized and quantum communication complexities of finding a maximum out-degree vertex are Theta(n log n), tilde{Theta}(n) and tilde{Theta}(sqrt{n}), respectively. Our upper bounds hold for all partitions of edges, and the lower bounds for a specific partition of the edges. To show the first bullet above, we show, perhaps surprisingly, that finding a source in a tournament is equivalent to the well-studied Clique vs. Independent Set (CIS) problem on undirected graphs. Our bounds for finding a source then follow from known bounds on the complexity of the CIS problem. In view of this equivalence, we can view the task of finding a king in a tournament to be a natural generalization of CIS. 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.
Paper Structure (22 sections, 27 theorems, 37 equations, 1 figure, 3 algorithms)

This paper contains 22 sections, 27 theorems, 37 equations, 1 figure, 3 algorithms.

Table of Contents

  1. Introduction
  2. Sketch of proofs
  3. Equivalence of source-finding and CIS
  4. Upper bounds
  5. Lower bounds
  6. Other results
  7. Preliminaries
  8. Query and Communication Complexity
  9. Decision tree rank and decision tree size
  10. Formal definitions of graph problems of interest
  11. Communication complexity of finding a source
  12. Communication complexity of $\mathsf{KING}$
  13. Upper bounds on communication complexity of $\mathsf{KING}_n$
  14. Lower bounds on communication complexity of $\mathsf{KING}_n$
  15. A class of tournaments
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $G$ be an $n$-vertex graph. Then, $\mathsf{D}^{\mathsf{cc}}(\mathsf{CIS}_G) = O(\log^2 n)$. Furthermore, there exists an $n$-vertex graph $G$ such that $\mathsf{D}^{\mathsf{cc}}(\mathsf{CIS}_G) = \widetilde{\Omega}(\log^2 n)$.

Figures (1)

  • Figure 1: Visual depiction of $G_{S, \sigma}$. For each $b \in \left\{0, 1, 2\right\}$, $S_b$ contains the vertices $\left\{i_b : i \in S\right\}$ and $T_b$ contains the vertices $\left\{i_b : i \notin S\right\}$. There are four types of edges (also see Definition \ref{['def: G S sigma']}): Edges of Type 1 are those within each $T_b \cup S_b$, here $i_b \rightarrow j_b$ iff $\sigma(i) > \sigma(j)$.Edges of Type 2 are those between $S_b$ and $T_{b'}$ for $b \neq b'$, here $i_b \rightarrow j_{b'}$ for all $b \neq b'$.Edges of Type 3 are those between $S_b$ and $S_{b'}$ for $b \neq b'$, here $i_b \rightarrow j_{b'}$ iff $b' = b+1~(\textnormal{mod } 3)$.Edges of Type 4 are those between $T_b$ and $T_{b'}$ for $b \neq b'$, here $i_b \rightarrow j_{b'}$ iff $b' = b+1~(\textnormal{mod } 3)$.

Theorems & Definitions (56)

  • Theorem 1.1: Yan91, GPW15
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 1.6
  • Definition 1.7: Permutation Maximum Finding
  • Theorem 1.9
  • Corollary 1.10
  • Theorem 1.11
  • ...and 46 more