When you come at the kings you best not miss
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A tournament is an orientation of a complete graph. We say that a vertex x in a tournament T⃗ controls another vertex y if there exists a directed path of length at most two from x to y. A vertex is called a king if it controls every vertex of the tournament. It is well known that every tournament has a king. We follow Shen, Sheng, and Wu (SIAM J. Comput., 2003) in investigating the query complexity of finding a king, that is, the number of arcs in T⃗ one has to know in order to surely identify at least one vertex as a king. The aforementioned authors showed that one always has to query at least Ω(n^4/3) arcs and provided a strategy that queries at most O(n^3/2). While this upper bound has not yet been improved for the original problem, Biswas et al. (Frontiers in Algorithmics, 2017) proved that with O(n^4/3) queries one can identify a semi-king, meaning a vertex which controls at least half of all vertices. Our contribution is a novel strategy which improves upon the number of controlled vertices: using O(n^4/3polylog n) queries, we can identify a (1/2+2/17)-king. To achieve this goal we use a novel structural result for tournaments.
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