Approximation threshold for nearest cluster-graph edit distance

Determine whether the nearest cluster-graph edit distance q^*(λ,μ) admits a polynomial-time approximation scheme, establish APX-hardness if it does not, and determine the best achievable approximation ratio relative to the strict factor-2 algorithm currently proved.

Background

The paper proves that deciding q^*(λ,μ)≤Q is strongly NP-complete and that exact evaluation admits no FPTAS unless P=NP. It also gives an O(n log n)-time alignment whose cost is strictly below 2q^*(λ,μ).

The reduction used for strong NP-completeness has zero approximation gap, so it neither establishes APX-hardness nor rules out a PTAS. Consequently, the precise approximation complexity and the optimal approximation ratio remain unresolved.