Theoretical and empirical evaluation of data reduction for exact Kemeny Rank Aggregation

Kemeny Rank Aggregation is a consensus finding problem important in many areas ranging from classical voting over web search and databases to bioinformatics. The underlying decision problem Kemeny Score is NP-complete even in case of four input rankings to be aggregated into a “median ranking”. We analyze efficient polynomial-time data reduction rules with provable performance bounds that allow us to find even all optimal median rankings. We show that our reduced instances contain at most candidates where d_a denotes the average Kendall’s tau distance between the input votes. On the theoretical side, this improves a corresponding result for a “partial problem kernel” from quadratic to linear size. In this context we provide a theoretical analysis of a commonly used data reduction. On the practical side, we provide experimental results with data based on web search and sport competitions, e.g., computing optimal median rankings for real-world instances with more than 100 candidates within milliseconds. Moreover, we perform experiments with randomly generated data based on two random distribution models for permutations.