Hardness of Median and Center in the Ulam Metric

2025-04-23

AI Summary

This paper investigates the median (minimizing the sum of Ulam distances) and center (minimizing the maximum Ulam distance) problems for rank aggregation under the Ulam metric, distinguishing between the continuous setting—where solutions may be arbitrary permutations—and the discrete setting—where solutions must belong to the input set of permutations. Via a carefully constructed reduction from Max-Cut and leveraging structural properties of the Ulam distance, we establish, for the first time, NP-hardness of both median and center computation in the continuous setting. In the discrete setting, we determine fine-grained complexity: we prove that the existing $ ilde{O}(n^2 L)$-time algorithms for median and center are conditionally optimal under the Strong Exponential Time Hypothesis (SETH), thereby fully resolving an open problem posed at APPROX’23.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Multiagent Systems: Mechanism Design

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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. $ullet$ 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]. $ullet$ 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.

Problem

• Investigates NP-hardness of median permutation in Ulam metric
• Resolves fine-grained complexity of discrete median and center
• Confirms NP-hardness of center permutation in Ulam metric

Innovation

• Proves NP-hardness of Ulam median via Max-Cut reduction
• Resolves fine-grained complexity of discrete Ulam problems
• Confirms conditional optimality of O(n²L) algorithm