# Query complexity of tournament solutions

Dey, P (2017) Query complexity of tournament solutions. In: 31st AAAI Conference on Artificial Intelligence, AAAI 2017, 4 - 10 February 2017, San Francisco, pp. 2992-2998.

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Official URL: https://doi.org/10.1016/j.engfailanal.2022.106442

## Abstract

A directed graph where there is exactly one edge between every pair of vertices is called a tournament. Finding the "best" set of vertices of a tournament is a well studied problem in social choice theory. A tournament solution takes a tournament as input and outputs a subset of vertices of the input tournament. However, in many applications, for example, choosing the best set of drugs from a given set of drugs, the edges of the tournament are given only implicitly and knowing the orientation of an edge is costly. In such scenarios, we would like to know the best set of vertices (according to some tournament solution) by "querying" as few edges as possible. We, in this paper, precisely study this problem for commonly used tournament solutions: given an oracle access to the edges of a tournament τ, find f(τ) by querying as few edges as possible, for a tournament solution f. We first show that the set of Condorcet non-losers in a tournament can be found by querying 2n-⌊log n⌋-2 edges only and this is tight in the sense that every algorithm for finding the set of Condorcet non-losers needs to query at least 2n-⌊log n⌋-2 edges in the worst case, where n is the number of vertices in the input tournament. We then move on to study other popular tournament solutions and show that any algorithm for finding the Copeland set, the Slater set, the Markov set, the bipartisan set, the uncovered set, the Banks set, and the top cycle must query Ω(n2) edges in the worst case. On the positive side, we are able to circumvent our strong query complexity lower bound results by proving that, if the size of the top cycle of the input tournament is at most k, then we can find all the tournament solutions mentioned above by querying O(nk + nlogn/log(1-1/k)) edges only.

Item Type: Conference Paper 31st AAAI Conference on Artificial Intelligence, AAAI 2017 AAAI press The copyright for this article belongs to AAAI press. Artificial intelligence, Input and outputs; Lower bounds; Oracle access; Positive sides; Query complexity; Social choice theory, Directed graphs Division of Electrical Sciences > Computer Science & Automation 22 Jul 2022 11:45 22 Jul 2022 11:45 https://eprints.iisc.ac.in/id/eprint/74741