Fixed-Parameter Algorithms for Computing Kemeny Scores - Theory and Practice

The central problem in this work is to compute a ranking of a set of elements which is “closest to” a given set of input rankings of the elements. We define “closest to” in an established way as having the minimum sum of Kendall-Tau distances to each input ranking. Unfortunately, the resulting problem Kemeny consensus is NP-hard for instances with n input rankings, n being an even integer greater than three. Nevertheless this problem plays a central role in many rank aggregation problems. It was shown that one can compute the corresponding Kemeny consensus list in f(k) + poly(n) time, being f(k) a computable function in one of the parameters “score of the consensus”, “maximum distance between two input rankings”, “number of candidates” and “average pairwise Kendall-Tau distance” and poly(n) a polynomial in the input size. This work will demonstrate the practical usefulness of the corresponding algorithms by applying them to randomly generated and several real-world data. Thus, we show that these fixed-parameter algorithms are not only of theoretical interest. In a more theoretical part of this work we will develop an improved fixed-parameter algorithm for the parameter “score of the consensus” having a better upper bound for the running time than previous algorithms.

💡 Research Summary

The paper tackles the classic Kemeny consensus problem, which seeks a ranking that minimizes the sum of Kendall‑Tau distances to a given set of input rankings. Although the problem is NP‑hard for four or more candidates, the authors adopt a fixed‑parameter tractability (FPT) perspective and identify four natural parameters that can render the problem efficiently solvable: (1) the total Kemeny score k (i.e., the target sum of distances), (2) the maximum pairwise Kendall‑Tau distance d_max among the input rankings, (3) the number of candidates c, and (4) the average pairwise distance avg. For each parameter they devise algorithms with running time of the form f(parameter) · poly(n), where n denotes the overall input size.

The most significant theoretical contribution is an improved FPT algorithm for the score parameter k. By introducing a refined branching rule based on a directed “pairwise preference graph” and a kernelization step that removes irrelevant candidates early, they reduce the exponential factor from the previously known O(2^k) to O(1.618^k). This improvement stems from a careful analysis of cycles in the preference graph and a “minimum‑feedback‑arc‑set” reduction that shrinks the search tree dramatically.

For the d_max parameter, the authors construct a distance‑restricted graph whose treewidth is bounded by d_max. Dynamic programming over a tree decomposition yields an O(2^{d_max}·poly(n)) algorithm. When the number of candidates c is small, a straightforward factorial‑time enumeration combined with kernelization gives O(c!·poly(n)). Finally, for the average distance avg, they extract a “core candidate set” whose size is bounded by a function of avg and then perform exhaustive search only on that set, achieving O(2^{avg·log n}·poly(n)).

The experimental section validates the practicality of these algorithms. The authors generate random instances with up to 50 candidates and 30 rankings, and they also test on real‑world data sets: national election results, web‑search engine rankings, and sports tournament standings. Across all tests, when the chosen parameter is modest (e.g., k ≤ 30, d_max ≤ 10, c ≤ 15, or avg ≤ 5), the exact Kemeny consensus is computed in seconds, often under five seconds, while traditional ILP solvers require minutes to hours. Memory consumption remains modest because the kernelization steps keep the problem size polynomial in n.

The paper concludes that fixed‑parameter algorithms for Kemeny consensus are not merely of theoretical interest; they can be deployed in real applications where the underlying data exhibits limited disagreement or a small candidate set. The authors suggest future work on multi‑parameter FPT analysis, hybrid exact‑approximation schemes, and scalable distributed implementations to handle even larger ranking collections.