The Nearest Neighbor Spearman Footrule Distance for Bucket, Interval, and Partial Orders

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. Recently, this approach has been generalized to bucket orders, which allow ties. In this work we further generalize and consider total, bucket, interval and partial orders. The disagreement between two orders is measured by the nearest neigbor Spearman footrule distance, which has not been studied so far. We show that the nearest neighbor Spearman footrule distance of two bucket orders and of a total and an interval order can be computed in linear time, whereas the computation is NP-complete and 6-approximable for a total and a partial order. Moreover, we establish the NP-completeness and the 4-approximability of the rank aggregation problem for bucket orders. This sharply contrasts the well-known efficient solution of this problem for total orders.