Clustering Permutations under the Ulam Metric: A Parameterized Complexity Study

TL;DR

This paper explores the parameterized complexity of rank aggregation problems under the Ulam metric, providing new algorithms and hardness results for various clustering variants.

Contribution

It introduces fixed-parameter algorithms for Ulam-based clustering problems and establishes complexity bounds, filling a gap in the understanding of these problems.

Findings

Ulam k-center is NP-hard for d=1 but FPT for k+d.

No polynomial kernel exists for k+d unless NP ⊆ coNP/poly.

Ulam k-median is W[1]-hard but admits an XP algorithm and a polynomial kernel.

Abstract

Rank aggregation seeks a representative permutation for a collection of rankings and plays a central role in areas such as social choice, information retrieval, and computational biology. Two fundamental aggregation tasks are the center and median problems, which minimize the maximum and the total distance to the input permutations, respectively. While these problems are well understood under Kendall's tau and related distances, their parameterized complexity under the Ulam metric, an edit-distance-based metric on permutations, has remained largely unexplored. In this work, we initiate a systematic study of the parameterized complexity of rank aggregation under the Ulam metric. We consider both the center and median problems, as well as their generalizations to the $k$-center and $k$-median clustering settings, parameterized by the number of centers $k$ and the distance budget $d$ (corresponding to the maximum distance for center variants and the total distance for median variants). Both problems are known to be NP-hard already for $k=1$. We show that the Ulam $k$-center problem remains NP-hard when $d=1$, but is fixed-parameter tractable when parameterized by $k + d$. Our algorithm is based on a novel local-search framework tailored to the non-local nature of Ulam distances. We complement this by proving that no polynomial kernel exists for the $k+d$ parameterization unless NP $\subseteq$ coNP/poly. For the Ulam $k$-median problem parameterized by the total distance $d$, we establish W[1]-hardness and provide an XP algorithm. We also provide a polynomial kernel for the parameter $k + d$, which in turn yields a fixed-parameter tractable algorithm.