Hardness of Approximation of Rank Aggregation on Ulam Metric

2026-08-29

AI Summary

研究了Ulam度量下排名聚合问题的近似难度，证明了对于Ulam中位数和中心问题，在多项式时间内难以获得接近最优解的结果。

Abstract

We study the approximability of rank aggregation under the Ulam metric. In the \emph{Ulam median} problem, the goal is to find a permutation minimizing the sum of its Ulam distances to the input permutations, while in the \emph{Ulam center} problem the objective is to minimize the maximum such distance. Both problems are known to be NP-hard, but no explicit approximation hardness was previously known. We prove that, for every $\varepsilon>0$, it is NP-hard to approximate either Ulam median or Ulam center within a factor of $51/50-\varepsilon$, even when the input consists of only four permutations. We further show that unless P = NP, neither problem admits a polynomial-time additive approximation scheme. The hardness result for Ulam median is established via a reduction from MAX-E3-LIN-2. The corresponding hardness for Ulam center is then obtained through a reduction from Ulam median.

Problem

• Ulam Metric
• Rank Aggregation
• Approximation Hardness
• Ulam Median
• Ulam Center

Innovation

• approximation hardness
• Ulam metric
• NP-hard
• polynomial-time additive approximation scheme
• reduction from MAX-E3-LIN-2