Hardness of Median and Center in the Ulam Metric

TL;DR

The paper addresses exact computation of median and center permutations under the Ulam metric for both continuous and discrete settings. It proves NP-hardness of the continuous Ulam median via a Max-Cut–based reduction and develops fine-grained, OV-based lower bounds showing that, in the discrete setting, the naive $ ilde{O}(n^2L)$ algorithm is essentially optimal under standard fine-grained hardness hypotheses. The work introduces a suite of gadgets (coordinate, vector) and balancing techniques to transfer orthogonality and Hamming-structure into Ulam distance, ultimately yielding near-tight conditional lower bounds for discrete center and median across parameter regimes. Together, these results sharply delineate the computational landscape for Ulam-based rank aggregation and illuminate the gap between Ulam and other string metrics like Hamming and edit distance, with implications for exact algorithm design in applications relying on consensus rankings.

Abstract

The classical rank aggregation problem seeks to combine a set X of n permutations into a single representative "consensus" permutation. In this paper, we investigate two fundamental rank aggregation tasks under the well-studied Ulam metric: computing a median permutation (which minimizes the sum of Ulam distances to X) and computing a center permutation (which minimizes the maximum Ulam distance to X) in two settings. $\bullet$ Continuous Setting: In the continuous setting, the median/center is allowed to be any permutation. It is known that computing a center in the Ulam metric is NP-hard and we add to this by showing that computing a median is NP-hard as well via a simple reduction from the Max-Cut problem. While this result may not be unexpected, it had remained elusive until now and confirms a speculation by Chakraborty, Das, and Krauthgamer [SODA '21]. $\bullet$ Discrete Setting: In the discrete setting, the median/center must be a permutation from the input set. We fully resolve the fine-grained complexity of the discrete median and discrete center problems under the Ulam metric, proving that the naive $\widetilde{O}(n^2 L)$-time algorithm (where L is the length of the permutation) is conditionally optimal. This resolves an open problem raised by Abboud, Bateni, Cohen-Addad, Karthik C. S., and Seddighin [APPROX '23]. Our reductions are inspired by the known fine-grained lower bounds for similarity measures, but we face and overcome several new highly technical challenges.

Figures (1)

• Figure 1: Illustration of the functions $f$ and $g$. $f(A)$ is obtained by taking $n$ copies of each normalized vector gadget in $A$ and then concatenating together the $n$ resulting strings. For $g(B)$, the order is flipped -- the $n$ normalized vector gadgets in $B$ are first concatenated together and then $n$ copies of the resulting string are concatenated.

Theorems & Definitions (52)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 4.1
• proof
• Definition 4.2
• Lemma 4.3
• proof
• Lemma 4.3
• Lemma 4.4