[Paper Review] Searching for Maximum Out-Degree Vertices in Tournaments

Gregory Gutin, George B. Mertzios+1 more|arXiv (Cornell University)|Jan 15, 2018

Artificial Intelligence in Games2 references3 citations

TL;DR

This paper establishes that finding a vertex of maximum out-degree in a tournament requires at least $\binom{n}{2} - O(n)$ binary inquiries in the worst case, significantly higher than the $\Theta(n^{3/2})$ cost for finding a king. The authors use an adversary argument on regular and almost-regular tournaments to prove tight lower bounds, showing that searching for maximum out-degree vertices is fundamentally more expensive than finding kings.

ABSTRACT

A vertex $x$ in a tournament $T$ is called a king if for every vertex $y$ of $T$ there is a directed path from $x$ to $y$ of length at most 2. It is not hard to show that every vertex of maximum out-degree in a tournament is a king. However, tournaments may have kings which are not vertices of maximum out-degree. A binary inquiry asks for the orientation of the edge between a pair of vertices and receives the answer. The cost of finding a king in an unknown tournament is the number of binary inquiries required to detect a king. For the cost of finding a king in a tournament, in the worst case, Shen, Sheng and Wu (SIAM J. Comput., 2003) proved a lower and upper bounds of $Ω(n^{4/3})$ and $O(n^{3/2})$, respectively. In contrast to their result, we prove that the cost of finding a vertex of maximum out-degree is ${n \choose 2} -O(n)$ in the worst case.

Motivation & Objective

• To determine the worst-case cost of identifying a vertex with maximum out-degree in a tournament using binary inquiries.
• To contrast the complexity of finding maximum out-degree vertices with that of finding kings, which is known to be $\Theta(n^{3/2})$.
• To establish tight lower bounds on the number of edge orientation queries required to correctly identify a maximum out-degree vertex.
• To demonstrate that restricting search to maximum out-degree vertices is not efficient for finding kings, as the cost regimes differ significantly.

Proposed method

• An adversary strategy is employed that maintains a fixed regular or almost-regular tournament structure throughout the query process.
• The adversary answers binary inquiries based on the actual arc orientations in the fixed tournament, but delays the algorithm’s ability to confirm a vertex as maximum out-degree.
• The adversary ensures that if insufficient queries are made, the algorithm cannot distinguish the true maximum out-degree vertex from others, by manipulating the remaining unknown arcs.
• Lower bounds are derived by analyzing the minimum number of incoming arcs required to be known for each non-target vertex to rule out their being maximum out-degree vertices.
• The proof uses case analysis based on the parity of $n$, distinguishing between odd $n$ (regular tournaments) and even $n$ (almost-regular tournaments).
• The sum of minimum required incoming degrees from all other vertices is used to derive the lower bound on query count.

Experimental results

Research questions

• RQ1 What is the worst-case number of binary inquiries required to identify a vertex of maximum out-degree in a tournament?
• RQ2 How does the query complexity of finding a maximum out-degree vertex compare to that of finding a king in a tournament?
• RQ3 Can a tight lower bound be established for the number of queries needed to identify a maximum out-degree vertex?
• RQ4 Is it feasible to restrict king-finding algorithms to only consider maximum out-degree vertices, given the query cost difference?

Key findings

• The worst-case cost of finding a maximum out-degree vertex in a tournament with $n$ vertices is at least $\binom{n}{2} - O(n)$ binary inquiries.
• For odd $n$, the lower bound is $\frac{(n-1)^2}{2}$ queries in the worst case.
• For even $n$, the lower bound is $\frac{(n-1)(n-2)}{2}$ queries in the worst case.
• These bounds are tight in the sense that they match the total number of edges minus a linear term, indicating near-complete knowledge of the tournament is required.
• The result shows that finding maximum out-degree vertices is significantly more expensive than finding kings, which can be done in $O(n^{3/2})$ queries.
• The adversary argument proves that no algorithm can correctly identify a maximum out-degree vertex with fewer than $\binom{n}{2} - O(n)$ queries in the worst case.