Almost Tight Approximation Hardness for Single-Source Directed k-Edge-Connectivity

TL;DR

The paper resolves long-standing questions about the approximability of the single-source directed $k$-edge-connected Steiner tree problem ($k$-DST) by deriving tight hardness results across several natural parameterizations. It introduces label-cover based reductions with padding and gadgetry (including padding arcs and a reverse $d$-ary arborescence) to control the numbers of terminals $|T|$ and the connectivity requirement $k$, preserving a strong completeness-soundness correspondence. The authors prove $oldmath{oldsymbol{rac{|T|}{ olinebreak oldsymbol{ ext{log}}|T|}}}$-hardness under NP $ eq$ ZPP, strengthenable to $oldmath{rac{|T|}{}}$ under SPCH, and establish hardness exponential in $k$ as well as $oldmath{igl(rac{k}{L}igr)^{L/4}}$ for $L$-layered graphs, nearly matching known approximation algorithms in several regimes. They also extend these inapproximability results to undirected variants such as $k$-GST and $k$-ST, providing a unified hardness landscape for survivable network design with Steiner vertices. Overall, the work substantially closes the gap between upper bounds and inapproximability for both directed and undirected single-source connectivity problems across multiple parameter regimes.

Abstract

In the $k$-connected directed Steiner tree problem ($k$-DST), we are given an $n$-vertex directed graph $G=(V,E)$ with edge costs, a connectivity requirement $k$, a root $r\in V$ and a set of terminals $T\subseteq V$. The goal is to find a minimum-cost subgraph $H\subseteq G$ that has $k$ internally disjoint paths from the root vertex $r$ to every terminal $t\in T$. In this paper, we show the approximation hardness of $k$-DST for various parameters, which thus close some long-standing open problems. - $Ω\left(|T|/\log |T|\right)$-approximation hardness, which holds under the standard assumption $\mathrm{NP}\neq \mathrm{ZPP}$. The inapproximability ratio is tightened to $Ω\left(|T|\right)$ under the Strongish Planted Clique Hypothesis [Manurangsi, Rubinstein and Schramm, ITCS 2021]. The latter hardness result matches the approximation ratio of $|T|$ obtained by a trivial approximation algorithm, thus closing the long-standing open problem. - $Ω\left(\sqrt{2}^k / k\right)$-approximation hardness for the general case of $k$-DST under the assumption $\mathrm{NP}\neq\mathrm{ZPP}$. This is the first hardness result known for survivable network design problems with an inapproximability ratio exponential in $k$. - $Ω\left((k/L)^{L/4}\right)$-approximation hardness for $k$-DST on $L$-layered graphs for $L\le O\left(\log n\right)$. This almost matches the approximation ratio of $O(k^{L-1}\cdot L \cdot \log |T|)$ achieving in $O\left(n^L\right)$-time due to Laekhanukit [ICALP`16].

Figures (11)

• Figure 1: A cover path $(r, u_{i}^a,v_{j}^b,v_j,t_{ij})$.
• Figure 2: An illegal path $(r, u_{i'}^a,v_{j}^b,v_j,t_{ij})$.
• Figure 3: $\Sigma=\left\{1,2,3\right\}$
• Figure 4: $\Sigma=\left\{1,2\right\}$
• Figure 5: $\pi_{u_iv_j}(x)= x\,\mathrm{mod}\, 3 + 1$

Theorems & Definitions (20)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7: Man2019
• Corollary 8
• Lemma 9: Folklore
• Claim 10