Ultrametric Violation Distance: Polynomial Kernel and FPT Algorithm

2026-08-10

AI Summary

This work addresses the problem of transforming a given distance matrix over $n$ points into an ultrametric by performing the minimum number of entry modifications, measured under the $\ell_0$ norm. Focusing on the number $k$ of violated ultrametric inequalities as the parameter, the paper presents the first polynomial kernel of size $O(k^2)$ for this problem and designs a single-exponential fixed-parameter tractable (FPT) algorithm with running time $9^k \cdot n^{O(1)}$. The proposed algorithm is asymptotically tight and represents a significant advance in the parameterized complexity of ultrametric fitting, markedly improving the efficiency of exact solutions for instances with few violations.

Abstract

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.

Problem

• Ultrametric Violation Distance
• ℓ₀-norm
• fixed-parameter tractability
• kernelization

Innovation

• Ultrametric Violation Distance
• polynomial kernel
• fixed-parameter tractability
• FPT algorithm
• ℓ₀-norm