Parameterized Inapproximability for Steiner Orientation by Gap Amplification

In the k\textnormal{-}\textsc{Steiner Orientation} problem, we are given a mixed graph, that is, with both directed and undirected edges, and a set of k terminal pairs. The goal is to find an orientation of the undirected edges that maximizes the number of terminal pairs for which there is a path from the source to the sink. The problem is known to be W[1]-hard when parameterized by k and hard to approximate up to some constant for FPT algorithms assuming Gap-ETH. On the other hand, no approximation factor better than O(k) is known.

 

We show that k\textnormal{-}\textsc{Steiner Orientation} is unlikely to admit an approximation algorithm with any constant factor, even within FPT running time. To obtain this result, we construct a self-reduction via a hashing-based gap amplification technique, which turns out useful even outside of the FPT paradigm. Precisely, we rule out any approximation factor of the form (\log k)^{o(1)} for FPT algorithms (assuming FPT \ne W[1]) and (\log n)^{o(1)} for purely polynomial-time algorithms (assuming that the class W[1] does not admit randomized FPT algorithms). This constitutes a novel inapproximability result for polynomial-time algorithms obtained via tools from the FPT theory.