A Tight Lower Bound For Steiner Orientation

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) is an element of 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, Kortsarz and Nutov [ESA'12, SIDMA'13] designed an XP algorithm running in n(O(k)) time for all k >= 1. Pilipczuk andWahlstrom [SODA'16] 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/log k)) algorithm for any computable function f. That is, the n(O(k)) algorithm of Cygan et al. is almost optimal.In this paper, we give a short and easy proof that the n(O(k)) algorithm of Cygan et al. is asymptotically optimal, even if the input graph has genus 1. Formally, we show that the 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 function f even if the underlying undirected graph has genus 1. We give a reduction from the Grid Tiling problem which has turned out to be very useful in proving W[1]-hardness of several problems on planar graphs. As a result of our work, the main remaining open question is whether Steiner Orientation admits the "square-root phenomenon" on planar graphs (graphs with genus 0): can one obtain an algorithm running in time f(k).n(O(root k)) for Planar Steiner Orientation, or does the lower bound of f(k).n(O(k))) also translate to planar graphs?