A Tight Algorithm for Strongly Connected Steiner Subgraph on Two Terminals with Demands

Given an edge-weighted directed graph G = ( V , E ) on n vertices and a set T = { t 1 , t 2 , … , t p } of p terminals, the objective of the Strongly Connected Steiner Subgraph ( p -SCSS) problem is to find an edge set H ⊆ E of minimum weight such that G [ H ] contains an t i → t j path for each 1 ≤ i ≠ j ≤ p . The p -SCSS problem is NP-hard, but Feldman and Ruhl [FOCS ’99; SICOMP ’06] gave a novel n O ( p ) time algorithm. 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- ( k 1 , k 2 ) 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 k 1 , k 2 ; the objective is to find a set of k 1 paths F 1 , F 2 , … , F k 1 from s ⇝ t and k 2 paths B 1 , B 2 , … , B k 2 from t ⇝ s such that ∑ e ∈ E ω ( e ) · ϕ ( e ) is minimized, where ϕ ( e ) = max { | { i ∈ [ k 1 ] : e ∈ F i } | , | { j ∈ [ k 2 ] : e ∈ B j } | } . For each k ≥ 1 , we show the following: The 2 -SCSS- ( k , 1 ) problem can be solved in time n O ( k ) . A matching lower bound for our algorithm: the 2 -SCSS- ( k , 1 ) problem does not have an f ( k ) · n o ( k ) time algorithm for any computable function f , unless the Exponential Time Hypothesis fails. Our algorithm for 2 -SCSS- ( k , 1 ) relies on a structural result regarding an 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- ( k 1 , k 2 ) problem if min { k 1 , k 2 } ≥ 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].