Almost Tight Approximation Hardness for Single-Source Directed k-Edge-Connectivity
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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∈ V and a set of terminals T⊆ V. The goal is to find a minimum-cost subgraph H⊆ G that has k internally disjoint paths from the root vertex r to every terminal t∈ T. In this paper, we show the approximation hardness of k-DST for various parameters, which thus close some long-standing open problems. - Ω(|T|/log |T|)-approximation hardness, which holds under the standard 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 / 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|) achieving in O(n^L)-time due to Laekhanukit [ICALP`16].
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