Ulam度量下秩聚合的近似难度
Hardness of Approximation of Rank Aggregation on Ulam Metric
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中文总结 AI 辅助
该研究针对Ulam度量下的Ulam中位数与Ulam中心两类秩聚合问题，证明了其51/50−ε因子的近似NP难度及无多项式时间加性近似方案的结论，并给出了相应的归约证明。
中文摘要 AI 辅助
我们研究了Ulam度量下秩聚合问题的近似性。在Ulam中位数问题中，目标是找到一个排列，使其与输入排列的Ulam距离之和最小；而在Ulam中心问题中，目标是最小化上述距离的最大值。已知这两个问题都是NP难的，但此前尚无明确的近似难度结论。我们证明，对任意ε>0，即使输入仅包含4个排列，将Ulam中位数或Ulam中心问题近似到51/50−ε的因子都是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加性近似方案。Ulam中位数的难度结果通过从MAX-E3-LIN-2归约得到，Ulam中心的对应难度则通过从Ulam中位数归约得出。
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 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.
发表机构
- Pennsylvania State University（宾夕法尼亚州立大学）
- National University of Singapore（新加坡国立大学）
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