How to Fix a Broken Metric: A Linear Kernel, Tight Bounds, and Tree Metrics

Abstract: Given a complete graph whose edge weights represent dissimilarities, Metric Violation Distance asks whether at most $k$ weights can be changed to form a metric. Motivated by metric repair for noisy data, the problem admits a polynomial-time $O(\log n)$-approximation due to Cohen-Addad, Fan, Lee and de Mesmay [SIAM J. Comput., 2025]. In the context of parameterized complexity, Fan, Gilbert, Raichel, Sonthalia and Van Buskirk [SWAT 2020] gave a $k<sup>{O(k)}n<sup>{O(1)}$-time algorithm. Fomin, Golovach and More [IPEC 2026] obtained an $O(k<sup>2)$ kernel and a single-exponential algorithm for the ultrametric case. They asked whether general metrics admit a single-exponential algorithm and a polynomial kernel, and whether the tree-metric analogue is fixed-parameter tractable. We answer all three questions. We give a $2<sup>{O(k)}n<sup>{O(1)}$-time algorithm and prove that, unless ETH fails, no $2<sup>{o(k)}n<sup>{O(1)}$-time algorithm exists, even when all input distances lie in ${1,2,3}$. We also give a kernel with at most $6k$ vertices. The kernel supports solution lifting and can precede any approximation algorithm. Combined with the $O(\log n)$-approximation of Cohen-Addad, Fan, Lee and de Mesmay, it yields an $O(\log \mathrm{OPT})$-approximation at no asymptotic cost in running time. Both results extend to an interval generalization in which each edge $e$ has an observed value $M_e$ and an admissible range $[A_e,B_e]$ within which it may be reassigned; the kernel then has $7k$ vertices. Finally, Tree Metric Violation Distance is fixed-parameter tractable and solvable in $k<sup>{O(k)}n<sup>{O(1)}$ time.