An ETH-Tight Algorithm for Bidirected Steiner Connectivity

In the Strongly Connected Steiner Subgraph problem, we are given an n-vertex digraph D, a weight function w: A(D) ↦ R⁺ on the arc set of D, and a set of k terminals Q⊆ V(D), and our objective is to find a strongly connected subgraph of D containing Q with minimum total weight. The problem is known to be W[1]-hard on general digraphs. However on bi-directed graphs (digraphs where, if uv is an arc then so is vu) with symmetric weight function w: A(D) ↦ R⁺ (i.e., w(uv) = w(vu) for any uv∈ A(D) ), Chitnis, Feldmann and Manurangsi [TALG 2021] showed that the problem is fixed parameter tractable (FPT) with running time 2O(k2)nO(1), where n is the input length. They also show that, unless the Exponential Time Hypothesis (ETH) fails, there is no algorithm for the problem on bi-directed graphs with running time 2 ^{o ( k )}n^{O ( 1 )}. They left the existence of a single-exponential in k time algorithm as an open problem. We resolve this question, by designing an algorithm for the problem running in time 2 ^{O ( k )}n^{O ( 1 )} that is asymptotically tight under ETH, thereby closing the gap between the upper and lower-bounds for this problem. Chitnis, Feldmann and Manurangsi [TALG 2021] showed that an optimum solution to this problem can always be described by a collection of trees, that are mapped to the input graph via homomorphisms, and glued together at the terminal vertices. This structural result allows us to design an algorithm via the combination of a Dreyfus-Wagner style dynamic programming algorithm and the notion of representative sets over linear matroids.