Optimal exponential base for the FPT algorithm

Determine the best possible base of the exponential dependence on k for the fixed-parameter algorithm for Metric Violation Distance.

Background

The randomized fixed-parameter algorithm for Metric Violation Distance succeeds with probability at least 2{-11k}, based on a random-separation construction in which the relevant red and blue edge sets have total size at most 11k. The resulting running time is single-exponential in k.

An ETH lower bound rules out a subexponential dependence 2{o(k)}, but the paper does not determine the optimal constant base in the exponent.

References

Similarly, the randomized FPT algorithm has success probability 2{-11k}, coming from the random-separation bound |O\cup Q|\le 11k. Although ETH rules out a subexponential dependence on k, the best possible base of the exponent is unknown.

— How to Fix a Broken Metric: A Linear Kernel, Tight Bounds, and Tree Metrics  (2610.07690 - Bandyapadhyay et al., 6 Oct 2026) in Section 8, Concluding Remarks