Ordering by weighted number of wins gives a good ranking for weighted tournaments

We consider the following simple algorithm for feedback arc set problem in weighted tournaments --- order the vertices by their weighted indegrees. We show that this algorithm has an approximation guarantee of 5 if the weights satisfy <i>probability constraints</i> (for any pair of vertices <i>u</i> and <i>v, w</i><inf><i>uv</i></inf> + w<inf><i>vu</i></inf> = 1). Special cases of feedback arc set problem in such weighted tournaments include feedback arc set problem in unweighted tournaments and rank aggregation. Finally, for any constant ε > 0, we exhibit an infinite family of (unweighted) tournaments for which the above algorithm (<i>irrespective</i> of how ties are broken) has an approximation ratio of 5 - ε.

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