Query complexity of tournament solutions

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 T, find f(T) by querying as few edges as possible, for a tournament solution f. We first study some popular tournament solutions and show that any deterministic 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. We also show similar lower bounds on the expected query complexity of these tournament solutions by any randomized algorithm. 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+nlog⁡(nk)log⁡(kk−1)) edges only.