Comparing and Aggregating Partial Orders with Kendall Tau Distances

Comparing and ranking information is an important topic in social and information sciences, and in particular on the web. Its objective is to measure the difference of the preferences of voters on a set of candidates and to compute a consensus ranking. Commonly, each voter provides a total order of all candidates. In this work we consider the generalization of total orders and bucket orders to partial orders and compare them by the nearest neighbor and the Hausdorff Kendall tau distances. First, we establish an O(n logn) algorithm for the computation of the nearest neighbor and the Hausdorff Kendall tau distances of two bucket orders. The computation of the nearest neighbor Kendall tau distance is NP-complete and 2-approximable for a total and a partial order. For the Hausdorff Kendall tau distance this problem is coNP-complete. Considering rank aggregation problems with partial orders, we establish a significant discrepancy between the two distances. For the nearest neighbor Kendall tau distance the problem is NP-complete even for two voters, whereas the Hausdorff Kendall tau distance problem is in Σp2 , but not in NP or coNP unless NP = coNP, even for four voters. However, both problems are known to be NP-complete for any even number of at least four total or bucket orders.