A New Approach for Approximating Directed Rooted Networks

TL;DR

This paper introduces the first approximation algorithm for the k-outconnected directed Steiner tree problem in graphs where most terminals have out-degree zero, achieving a ratio dependent on the size of the non-terminal set.

Contribution

It presents a novel randomized approximation algorithm tailored for a specific class of directed graphs, addressing a previously hard problem.

Findings

Achieves an approximation ratio of O(k|S| log |T|).

Runs in polynomial time with high probability.

Addresses a gap in algorithms for k-DST in specific graph classes.

Abstract

We consider the k-outconnected directed Steiner tree problem (k-DST). Given a directed edge-weighted graph $G=(V,E,w)$, where $V=\{r\}\cup S \cup T$, and an integer $k$, the goal is to find a minimum cost subgraph of $G$ in which there are $k$ edge-disjoint $rt$-paths for every terminal $t\in T$. The problem is know to be NP-hard. Furthermore, the question on whether a polynomial time, subpolynomial approximation algorithm exists for $k$-DST was answered negatively by Grandoni et al. (2018), by proving an approximation hardness of $\Omega (|T|/\log |T|)$ under $NP\neq ZPP$. Inspired by modern day applications, we focus on developing efficient algorithms for $k$-DST in graphs where terminals have out-degree $0$, and furthermore constitute the vast majority in the graph. We provide the first approximation algorithm for $k$-DST on such graphs, in which the approximation ratio depends (primarily) on the size of $S$. We present a randomized algorithm that finds a solution of weight at most $\mathcal O(k|S|\log |T|)$ times the optimal weight, and with high probability runs in polynomial time.