A New Approach for Approximating Directed Rooted Networks

TL;DR

This work addresses the $k$-outconnected directed Steiner tree problem ($k$-DST) in graphs where terminals tend to dominate and have out-degree $0$. The authors introduce a novel combination of connectivity augmentation, implicit hitting-set rounding, and a strict-core framework for $T$-intersecting families, culminating in a randomized Las Vegas algorithm that achieves an $O(k|S|\, ext{log}|T|)$-approximation in polynomial time w.h.p. The approach relies on iterative augmentation, an implicit-hitting-set oracle, and a new auxiliary graph construction to ensure that covering strict cores suffices to cover all cores, enabling efficient rounding. The results show that the approximation depends primarily on the Steiner set size $|S|$ and remains favorable when $|S|=O(|T|^{1/d})$ or polylogarithmic in $|T|$, making it practical for terminal-heavy networks. Overall, the paper advances practical design of robust, directed networks in applications like multicast routing and healthcare information distribution by providing the first strong approximation for this regime with provable performance guarantees.

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 $Ω(|T|/\log |T|)$ 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.

Figures (1)

• Figure 1: Constructing an Auxiliary Graph $H_t$

Theorems & Definitions (32)

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• definition 1
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• proof
• definition 2
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