When you come at the King you best not miss

Lachish, Oded and Reidl, Felix and Trehan, Chhaya (2022) When you come at the King you best not miss. 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022) 250 , 25:1-25:12. ISSN 1868-8969.

Abstract

A tournament is an orientation of a complete graph. We say that a vertex x in a tournament ⃗ T 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 [ 8] in investigating the query complexity of finding a king, that is, the number of arcs in ⃗ T 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(n4/3) arcs and provided a strategy that queries at most O(n3/2). While this upper bound has not yet been improved for the original problem, Biswas et al. [3] proved that with O(n4/3) 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(n4/3 polylog 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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