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

Abstract

Given an edge-weighted directed graph $$G=(V,E)$$G=(V,E) on n vertices and a set $$T=\{t_1, t_2, \ldots t_p\}$$T={t1,t2,?,tp} of p terminals, the objective of the Strongly Connected Steiner Subgraph (p-SCSS) problem is to find an edge set $$H\subseteq E$$H⊆E of minimum weight such that G[H] contains an $$t_{i}\rightarrow t_j$$ti?tj path for each $$1\le i\ne j\le p$$1≤i?j≤p. The p-SCSS problem is NP-hard, but Feldman and Ruhl [FOCS '99; SICOMP '06] gave a novel $$n^{O(p)}$$nO(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 $$2$$2-SCSS-$$(k_1, k_2)$$(k1,k2) problem is defined as follows: given an edge-weighted directed graph $$G=(V,E)$$G=(V,E) with weight function $$\omega : E\rightarrow {\mathbb {R}}^{\ge 0}$$?:E?R?0, two terminal vertices s, t, and integers $$k_1, k_2$$k1,k2; the objective is to find a set of $$k_1$$k1 paths $$F_1, F_2, \ldots F_{k_1}$$F1,F2,?,Fk1 from $$s\leadsto t$$s?t and $$k_2$$k2 paths $$B_1, B_2, \ldots B_{k_2}$$B1,B2,?,Bk2 from $$t\leadsto s$$t?s such that $$\sum _{e\in E} \omega (e)\cdot \phi (e)$$?e?E?(e)·?(e) is minimized, where $$\phi (e)= \max \Big \{|\{i\in [k_1] : e\in F_i\}| , |\{j\in [k_2] : e\in B_j\}|\Big \}$$?(e)=max{|{i?[k1]:e?Fi}|,|{j?[k2]:e?Bj}|}. For each $$k\ge 1$$k?1, we show the following:The $$2$$2-SCSS-$$(k,1)$$(k,1) problem can be solved in time $$n^{O(k)}$$nO(k).A matching lower bound for our algorithm: the $$2$$2-SCSS-$$(k,1)$$(k,1) problem does not have an $$f(k)\cdot n^{o(k)}$$f(k)·no(k) time algorithm for any computable function f, unless the Exponential Time Hypothesis fails. Our algorithm for $$2$$2-SCSS-$$(k,1)$$(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 $$2$$2-SCSS-$$(k_1, k_2)$$(k1,k2) problem if $$\min \{k_1, k_2\}\ge 2$$min{k1,k2}?2. Therefore $$2$$2-SCSS-$$(k,1)$$(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].