AI 中文总结
本文研究ℓ₀范数下的超度量违反距离问题，证明其存在含O(k²)个点的核，且提出运行时间为9ᵏ·n^O(1)的单指数FPT算法，推进了该问题的核化与FPT研究。
AI 中文摘要
在超度量违反距离问题中，给定n个点之间的一组距离，目标是修改最少数量的距离，使得到的集合形成有效的超度量。换句话说，任务是将超度量拟合到给定数据，拟合质量由误差的ℓ₀范数衡量。虽然该问题在ℓ_∞和ℓ₁范数下的变体已得到充分研究，但ℓ₀范数下超度量违反距离的复杂性直到最近仍未被深入探索。这一情况因Cohen-Addad、Fan、Lee和Mesmay（2022年FOCS会议）的工作而改变，他们提出了一种常数因子近似算法。Charikar和Gao（2024年SODA会议）以及An、Kao、Lee和Lee（2025年FOCS会议）的后续工作在近似算法方面取得了重大进展。本文从核化和固定参数可处理性（FPT）的角度对超度量违反距离展开系统研究。根据Fan、Gilbert、Raichel、Sonthalia和Van Buskirk（2020年SWAT会议）的研究，已知当参数为违反距离的数量k时，该问题是FPT的。本文证明该问题存在具有O(k²)个点的核，此外，还提出了一种单指数时间算法，运行时间为9ᵏ·n^O(1)，该算法在渐近意义上是紧的。
英文摘要
In the Ultrametric Violation Distance problem, we are given a set of distances between $n$ points, and the goal is to modify the minimum number of distances so that the resulting set forms a valid ultrametric. In other words, the task is to fit an ultrametric to the given data, where the quality of the fit is measured by the $\ell_0$-norm of the error. While variants of this problem under the $\ell_\infty$ and $\ell_1$-norms have been well studied, the complexity of Ultrametric Violation Distance under the $\ell_0$-norm remained largely unexplored until recently. This changed with the work of Cohen-Addad, Fan, Lee, and Mesmay [FOCS 2022], who introduced a constant-factor approximation algorithm. Significant further progress on approximation algorithms was made in subsequent work by Charikar and Gao [SODA 2024], and by An, Kao, Lee, and Lee [FOCS 2025]. In this paper, we initiate a systematic study of Ultrametric Violation Distance from the perspectives of kernelization and fixed-parameter tractability (FPT). By the work of Fan, Gilbert, Raichel, Sonthalia, and Van Buskirk [SWAT 2020], the problem is known to be FPT when parameterized by the number of violated distances $k$. We show that the problem admits a kernel with $\mathcal{O}(k^2)$ points. Additionally, we present a single-exponential-time algorithm with running time $9^k \cdot n^{\mathcal{O}(1)}$, which is asymptotically tight.
