A tight algorithm for strongly connected steiner subgraph on two terminals with demands (extended abstract)

Given an edge-weighted directed graph G = (V, E) on n vertices and a set T = {t1, t2,… tp} of p terminals, the objective of the Strongly Connected Steiner Subgraph (SCSS) problem is to find an edge set H ⊆ E of minimum weight such that G[H] contains a ti → tj path for each 1 ≤ i _= j ≤ p. The problem is NP-hard, but Feldman and Ruhl [FOCS’99; SICOMP’06] gave a novel nO(p) algorithm for the p-SCSS problem. In this paper, we investigate the computational complexity of a variant of 2-SCSS where we have demands for the number of paths between each terminal pair. Formally, the 2-SCSS-(k1, k2) problem is defined as follows: given an edge-weighted directed graph G = (V, E) with weight function ω: E → R≥0, two terminal vertices s, t, and integers k₁, k₂; the objective is to find a set of k1 paths F₁, F₂,…, F_k1 from s → t and k₂ paths B₁, B₂,…, B_k2 from t → s such that ∑_e∈E ω(e)·φ(e) is minimized, where φ(e) = max{|{i: i ∈ [k₁], e ∈ Fi}|; |{j: j ∈ [k₂], e ∈ B_j}|}. For each k ≥ 1, we show the following:-The 2-SCSS-(k, 1) problem can be solved in n^O(k) time.-A matching lower bound for our algorithm: the 2-SCSS-(k, 1) problem does not have an f(k) · n^o(k) algorithm for any computable function f, unless the Exponential Time Hypothesis (ETH) fails. Our algorithm for 2-SCSS-(k, 1) relies on a structural result regarding the optimal solution followed by using the idea of a “token game” similar to that of Feldman and Ruhl. We show with an example that the structural result does not hold for the 2-SCSS-(k₁, k₂) problem if min{k₁, k₂} ≥ 2. Therefore 2-SCSS-(k, 1) is the most general problem one can attempt to solve with our techniques. To obtain the lower bound matching the algorithm, we reduce from a special variant of the Grid Tiling problem introduced by Marx [FOCS’07; ICALP’12].