Crossings and Permutations

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We investigate crossing minimization problems for a set of permutations, where a crossing expresses a disarrangement between elements. The goal is a common permutation π * which minimizes the number of crossings. This is known as the Kemeny optimal aggregation problem minimizing the Kendall-τ distance. Recent interest into this problem comes from application to meta-search and spam reduction on the Web.

This rank aggregation problem can be phrased as a one-sided two-layer crossing minimization problem for an edge coloured bipartite graph, where crossings are counted only for monochromatic edges.

Here we introduce the max version of the crossing minimization problem, which attempts to minimize the discrimination against any permutation. We show the NP-hardness of the common and the max version for k ≥ 4 permutations (and k even), and establish a 2-2/k and a 2-approximation, respectively. For two permutations crossing minimization is solved by inspecting the drawings, whereas it remains open for three permutations.