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

Authors Chao Liao, Qingyun Chen, Bundit Laekhanukit, Yuhao Zhang

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Author Details

Chao Liao
  • Shanghai Jiao Tong University, China
Qingyun Chen
  • University of California, Merced, CA, USA
Bundit Laekhanukit
  • Shanghai University of Finance and Economics, China
Yuhao Zhang
  • Shanghai Jiao Tong University, China

Funding

  • Chen, Qingyun: Qingyun Chen is supported in part by NSF grants CCF-2121745 and CCF-1844939.
  • Laekhanukit, Bundit: Bundit Laekhanukit is supported by the 1000-talent plan award by the Chinese government, supported by Science and Technology Innovation 2030 –- "New Generation of Artificial Intelligence" Major Project No.(2018AAA0100903), NSFC grant 61932002, Program for Innovative Research Team of Shanghai University of Finance and Economics (IRTSHUFE) and the Fundamental Research Funds for the Central Universities.
  • Zhang, Yuhao: Yuhao Zhang is supported by National Natural Science Foundation of China (NSFC) under Grant No. 62102251.

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Chao Liao, Qingyun Chen, Bundit Laekhanukit, and Yuhao Zhang. Almost Tight Approximation Hardness for Single-Source Directed k-Edge-Connectivity. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 89:1-89:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ICALP.2022.89

Abstract

In the k-outconnected 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 ∈ V and a set of terminals T ⊆ V. The goal is to find a minimum-cost subgraph H ⊆ G that has k edge-disjoint paths from the root vertex r to every terminal t ∈ T. The problem is NP-hard, and inapproximability results are known in several parameters, e.g., hardness in terms of n: log^{2-ε}n-hardness for k = 1 [Halperin and Krauthgamer, STOC'03], 2^{log^{1-ε}n}-hardness for general case [Cheriyan, Laekhanukit, Naves and Vetta, SODA'12], hardness in terms of k [Cheriyan et al., SODA'12; Laekhanukit, SODA'14; Manurangsi, IPL'19] and hardness in terms of |T| [Laekhanukit, SODA'14]. In this paper, we show the approximation hardness of k-DST for various parameters. - Ω(|T|/log |T|)-approximation hardness, which holds under the standard complexity assumption NP≠ ZPP. The inapproximability ratio is tightened to Ω(|T|) 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(2^{k/2} / k)-approximation hardness for the general case of k-DST under the assumption NP≠ZPP. This is the first hardness result known for survivable network design problems with an inapproximability ratio exponential in k. - Ω((k/L)^{L/4})-approximation hardness for k-DST on L-layered graphs for L ≤ O(log n). This almost matches the approximation ratio of O(k^{L-1}⋅ L ⋅ log |T|) achieved in O(n^L)-time due to Laekhanukit [ICALP'16]. We further extend our hardness results in terms of |T| to the undirected cases of k-DST, namely the single-source k-vertex-connected Steiner tree and the k-edge-connected group Steiner tree problems. Thus, we obtain Ω(|T|/log |T|) and Ω(|T|) approximation hardness for both problems under the assumption NP≠ ZPP and the Strongish Planted Clique Hypothesis, respectively. This again matches the upper bound obtained by trivial algorithms.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Approximation algorithms
  • Mathematics of computing → Paths and connectivity problems
Keywords
  • Directed Steiner Tree
  • Hardness of Approximation
  • Fault-Tolerant and Survivable Network Design

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