Inapproximability within W[1]: the case of Steiner Orientation
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In the k-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 better than O(k) is known. We show that k-Steiner Orientation admits no sublogarithmic approximation algorithm, even with a parameterized running time, assuming W[1] FPT. To obtain this result, we reduce the problem to itself via a hashing-based gap amplification technique, which turns out useful even outside of the FPT paradigm. Precisely, we rule out any approximation ratio of the form ( k)^o(1) for parameterized algorithms and ( n)^o(1) for purely polynomial running time, under the same assumption. This constitutes a novel inapproximability result for polynomial time algorithms obtained via tools from the FPT theory. Moreover, we prove k-Steiner Orientation to be W[1]-complete, what provides an example of a natural approximation task that is complete in a parameterized complexity class. Finally, we apply our technique to the maximization version of Directed Multicut and obtain a simple proof that the problem admits no FPT approximation with ratio O(k^1/2 - ϵ) (assuming W[1] FPT) and no polynomial-time approximation with ratio O(m^1/2 - ϵ) (assuming NP ⊆ co-RP).
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