An ETH-Tight Algorithm for Bidirected Steiner Connectivity

Daniel Lokshtanov, Pranabendu Misra, Fahad Panolan, Saket Saurabh, Meirav Zehavi

Abstract: In the Strongly Connected Steiner Subgraph problem, we are given an n-vertex digraph D, a weight function \(w:A(D)\mapsto {\mathbb {R}}^+\) on the arc set of D, and a set of k terminals \(Q\subseteq V(D)\), 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 \(w:A(D)\mapsto {\mathbb {R}}^+\) (i.e., \(w(uv)=w(vu)\) for any \(uv\in A(D)\)), Chitnis, Feldmann and Manurangsi [TALG 2021] showed that the problem is fixed parameter tractable (FPT) with running time \(2^{\mathcal {O}({k^2})}n^{\mathcal {O}(1)}\), 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 \(2^{o(k)}n^{\mathcal {O}(1)}\). 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 \(2^{\mathcal {O}(k)}n^{\mathcal {O}(1)}\) that is asymptotically tight under ETH, thereby closing the gap between the upper and lower-bounds for this problem.