关于在锦标赛中寻找国王的通信复杂度
On the communication complexity of finding a king in a tournament
AI总结：
本文研究边被划分给两方的锦标赛中寻找源点、国王和最大出度顶点的通信复杂度，给出确定性、随机化和量子复杂度的上下界，并揭示寻找源点与 CIS 问题的等价性。
摘要：
锦标赛是一种完全有向图。锦标赛中的国王是指一个顶点 v，使得从 v 出发经过长度至多 2 的路径可以到达其他所有顶点。众所周知，每个锦标赛至少有一个国王，其中一个国王是最大出度顶点。在查询复杂度背景下，寻找锦标赛中的国王、最大出度顶点和源点这三项任务已经得到了相对充分的研究。我们研究这些任务的通信复杂度，其中边被划分给两个参与者。以下是针对 n 个顶点锦标赛的主要结果：\n1) 判断是否存在源点的确定性通信复杂度为 tilde{Theta}(log^2 n)。\n2) 寻找国王的确定性通信复杂度和随机化通信复杂度均为 Theta(n)。量子通信复杂度为 tilde{Theta}(sqrt{n})。\n3) 寻找最大出度顶点的确定性、随机化和量子通信复杂度分别为 Theta(n log n)、tilde{Theta}(n) 和 tilde{Theta}(sqrt{n})。\n我们的上界对所有边的划分方式都成立，下界则针对某种特定的边划分方式。为了证明上述第一点，我们证明了一个或许令人惊讶的结论：在锦标赛中寻找源点等价于无向图上已被充分研究的 Clique vs. Independent Set（CIS）问题。于是，我们关于寻找源点的界可由 CIS 问题复杂度的已知界直接推出。鉴于这一等价性，我们可以把在锦标赛中寻找国王的任务视为 CIS 的自然推广。\n我们的一个下界使用了基于 fooling-set 的论证，其余所有下界都来自精心构造的 Set-Disjointness 归约。
英文摘要：
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.
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