Constant-additive-error approximation complexity for information bottleneck

Determine whether the optimal values of the constrained information bottleneck function $IB(P_{XY},R)$ and the Lagrangian information bottleneck function $L_{\beta}(P_{XY})$ can be approximated within a constant additive error in polynomial time for arbitrary finite-alphabet joint distributions.

Background

The paper establishes NP-hardness and inverse-polynomial additive inapproximability for both the constrained information bottleneck value $IB(P_{XY},R)$ and the Lagrangian value $L_{\beta}(P_{XY})$ on explicitly specified rational finite-alphabet distributions. In particular, the proved lower bounds rule out polynomial-time approximation within errors of order $|X|^{-3/2}$ unless $\mathrm{P}=\mathrm{NP}$.

The authors leave unresolved whether this computational barrier persists for approximation within a constant additive error, which would provide a substantially stronger hardness result than the inverse-polynomial bounds proved in the paper. The question concerns the general information bottleneck problem without the structural assumptions used in known exactly solvable cases.