Abstrakt
In the Steiner Orientation problem, the input is a mixed graph G (it has both directed and undirected edges) and a set of k terminal pairs T. The question is whether we can orient the undirected edges in a way such that there is a directed s-t path for each terminal pair (s,t)T.
Arkin and Hassin [DAM'02] showed that the Steiner Orientation problem is NP-complete. They also gave a polynomial time algorithm for the special case when k=2.
From the viewpoint of exact algorithms, Cygan et al.[ESA'12, SIDMA'13] designed an XP algorithm running in n^O(k) time for all k>=1. Pilipczuk and Wahlstrom [SODA'16, TOCT'18] showed that the Steiner Orientation problem is W[1]-hard parameterized by k.
As a byproduct of their reduction, they were able to show that under the Exponential Time Hypothesis (ETH) of Impagliazzo, Paturi and Zane [JCSS'01] the Steiner Orientation problem does not admit an f(k)*n^o(k/logk) algorithm for any computable function f. In this paper, we give a short and easy proof that the n^O(k) algorithm of Cygan etal. is asymptotically optimal, even if the input graph is planar.
Formally, we show that the Planar Steiner Orientation problem is W[1]-hard parameterized by the number k of terminal pairs, and, under ETH, cannot be solved in f(k)n^o(k) time for any computable function f.