A Tight Algorithm for Strongly Connected Steiner Subgraph on Two Terminals with Demands (Extended Abstract)

Abstract

Given an edge-weighted directed graph $G=(V,E)$ on $n$ vertices and a set $T=\{t_1, t_2, \ldots t_p\}$ of $p$ terminals, the objective of the Strongly Connected Steiner Subgraph (SCSS) problem is to find an edge set $H\subseteq E$ of minimum weight such that $G[H]$ contains a $t_{i}\rightarrow t_j$ path for each $1\le i\ne j\le p$. The problem is NP-hard, but Feldman and Ruhl [FOCS ’99; SICOMP ’06] gave a novel $n^{O(p)}$ 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-$(k_1, k_2)$ problem is defined as follows: given an edge-weighted directed graph $G=(V,E)$ with weight function $\omega : E\rightarrow \mathbb {R}_{\ge 0}$, two terminal vertices $s, t$, and integers $k_1, k_2$ ; the objective is to find a set of $k_1$ paths $F_1, F_2, \ldots , F_{k_1}$ from $s\leadsto t$ and $k_2$ paths $B_1, B_2, \ldots , B_{k_2}$ from $t\leadsto s$ such that $\sum _{e\in E} \omega (e)\cdot \phi (e)$ is minimized, where $\phi (e)= \max \Big \{|\{i : i\in [k_1], e\in F_i\}|\ ;\ |\{j : j\in [k_2], e\in B_j\}|\Big \}$. For each $k\ge 1$, 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 $f(k)\cdot n^{o(k)}$ 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-$(k_1, k_2)$ problem if $\min \{k_1, k_2\}\ge 2$. 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].

A full version of the paper is available on arXiv.org.

Chitnis, Esfandiari, Hajiaghayi, and Seddighin—Supported in part by NSF CAREER award 1053605, NSF grant CCF-1161626, ONR YIP award N000141110662, DARPA/AFOSR grant FA9550-12-1-0423.

Guy Kortsarz—Supported by NSF grant 1218620.