A King for Almost Everyone

Abstract

A tournament is an orientation of a complete graph. A vertex $v$ controls a vertex $u\ne v$ if there is a directed path of length at most two from $v$ to $u$, and a $\delta$-king is one that controls at least $\delta (n-1)$ vertices. In the query complexity model, each query reveals the orientation of an edge. Shen, Sheng and Wu [SICOMP'03] established an $\Omega (n^{4/3})$ lower bound on the deterministic query complexity of finding a $1$-king, and a non-matching upper bound of $O(n^{3/2})$. Neither bound has been improved in over two decades, despite substantial progress on variants of the problem. For $\delta$-kings with $\delta <1$, Biswas, Jayapaul, Raman and Satti [Discrete Applied Mathematics'22] showed that $O(n^{4/3})$ queries suffice to find a $\frac{1}{2}$-king. Lachish, Reidl and Trehan [FSTTCS'22] improved this guarantee to $\frac{1}{2}+\frac{2}{17}$ using $\tilde{O}(n^{4/3})$ queries, and asked whether a larger fraction could be achieved within the same query bound. We answer this question affirmatively for every fixed fraction below one: for every fixed $\epsilon \in (0,1)$, we give a deterministic algorithm that finds a $(1-\epsilon )$-king using $O(n^{4/3})$ queries. The design of our algorithm is informed by a structural analysis combining a sparse regularity lemma due to Allen, Kohayakawa, Mota and Parente [Electronic Journal of Combinatorics'14] with a new sampling argument based on a reachability theorem of Afek, Gafni and Linial [arXiv'19].

arXiv comment13 pages, 1 figure