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arXiv · 2610.40521

Directed Cut-Query Complexity of Finding a King

Ziad Ismaili Alaoui·Oded Lachish·Nikhil S. Mande
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Abstract

A king in a tournament is a vertex from which every other vertex is reachable by a directed path of length at most two. We study the deterministic query complexity of finding a king when a query specifies a set of vertices and returns the number of edges directed from that set to its complement. We prove that this complexity is $Θ(n)$ for $n$-vertex tournaments. To show this, we first show that every correct deterministic algorithm in this model must output a maximum-outdegree vertex, even though a king need not have maximum outdegree. We then give an adversary whose answers, after too few queries, remain consistent with a tournament in which the algorithm's output does not have maximum outdegree.

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Ziad Ismaili Alaoui, Oded Lachish, Nikhil S. Mande. 2026-10-29. Directed Cut-Query Complexity of Finding a King. https://arxiv.org/abs/2610.40521

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