Ulam 排名聚合在四个排名情况下难以近似
Ulam Rank Aggregation Is Hard to Approximate for Four Rankings
- Pennsylvania State University（宾夕法尼亚州立大学）
- National University of Singapore（新加坡国立大学）
- Rutgers University（罗格斯大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明在 Ulam 度量下，即使输入仅四个排名，Ulam 中位数和中心问题在因子 35/34-ε 内近似是 NP-难的，并引入 CSP 归约框架，结果在输入数量上最优。
AI中文摘要：
我们研究了 Ulam 度量下排名聚合的可近似性。在 Ulam 中位数问题中，目标是找到一个排名（排列），使其与输入排名的 Ulam 距离之和最小；而在 Ulam 中心问题中，目标是最小化这些距离的最大值。我们证明，对于每个 $0<\varepsilon< 1/34$，即使输入仅包含四个排名，在因子 $35/34-\varepsilon$ 内近似 Ulam 中位数或 Ulam 中心都是 $\mathrm{NP}$-难的。我们进一步表明，除非 P = NP，否则这两个问题都不存在多项式时间的加法近似方案。在我们之前的工作中，只有这两个问题的精确版本已知是 $\mathrm{NP}$-难的，而且仅在输入排名数量无界的情况下 [Fischer et al., ESA'25 和 Bachmaier et al., J. of Discrete Algorithms'15]。此外，我们的不可近似性结果在输入排名数量方面是最优的，因为对于三个输入，已知是多项式时间可解的 [Chakraborty, Das, Krauthgamer, SODA'21]。在此过程中，我们引入了一个新的通用框架，用于将布尔约束满足问题（CSP）归约到仅四个输入的 Ulam 中位数。作为归约框架的一个具体实例，我们获得了不可近似性结果。Ulam 中心的相应难度来自 Ulam 中位数的归约。
英文摘要：
We study the approximability of rank aggregation under the Ulam metric. In the \emph{Ulam median} problem, the goal is to find a ranking (permutation) minimizing the sum of its Ulam distances to the input rankings, while in the \emph{Ulam center} problem, the objective is to minimize the maximum such distance. We prove that, for every $0<\varepsilon< 1/34$, it is $\mathrm{NP}$-hard to approximate either Ulam median or Ulam center within a factor of $35/34-\varepsilon$, even when the input consists of only four rankings. We further show that unless P = NP, neither problem admits a polynomial-time additive approximation scheme. Prior to our work, only the exact versions of both problems were known to be $\mathrm{NP}$-hard, and that too only when the number of input rankings is unbounded [Fischer et al., ESA'25 and Bachmaier et al., J. of Discrete Algorithms'15]. Furthermore, our inapproximability results are optimal in terms of the number of input rankings since for three inputs it is already known to be polynomial-time solvable [Chakraborty, Das, Krauthgamer, SODA'21]. En route, we introduce a new general framework for reducing Boolean constraint satisfaction problems (CSP) to the Ulam median with only four inputs. As a specific instantiation of the reduction framework, we obtain our hardness-of-approximation results. The corresponding hardness for the Ulam center follows from a reduction from the Ulam median.
