Constant-factor approximation for Metric Violation Distance

Determine whether Metric Violation Distance admits a constant-factor approximation in polynomial time.

Background

Metric Violation Distance asks for the minimum number of edge lengths in a complete weighted graph that must be modified so that the resulting distances satisfy the metric inequalities. The paper provides a linear solution-lifting kernel and shows that the constant-factor approximation question can be restricted to instances with a number of vertices linear in the optimum.

Despite these reductions and existing logarithmic-factor approximation algorithms, the existence of a polynomial-time constant-factor approximation remains unresolved.