Single-exponential algorithm or polynomial kernel for Tree Metric Violation Distance

Establish whether Tree Metric Violation Distance admits a \(2^{\mathcal{O}(k)}n^{\mathcal{O}(1)}\)-time algorithm or a polynomial kernel.

Background

Tree Metric Violation Distance asks whether at most k pairwise distances can be changed so that the resulting distance function is a positive tree metric. The paper establishes fixed-parameter tractability with running time k^{\mathcal{O}(k)}n^{\mathcal{O}(1)}, using bounded search and an oracle that produces obstructions of size \mathcal{O}(k^2).

The paper identifies two technical bottlenecks: the quadratic-size obstruction generated by the partial-completion procedure and the enumeration of k^{\mathcal{O}(k)} tree topologies. It does not resolve whether the problem can be solved in single-exponential parameter dependence or admits a polynomial kernel.