Parameterized Approximation Algorithms for Bidirected Steiner Network Problems

The Directed Steiner Network (DSN) problem takes as input a directed edge-weighted graph $G=(V,E)$ and a set $\mathcal{D}\subseteq V\times V$ of $k$ demand pairs. The aim is to compute the cheapest network $N\subseteq G$ for which there is an $s\to t$ path for each $(s,t)\in\mathcal{D}$. It is known that this problem is notoriously hard as there is no $k^{1/4-o(1)}$-approximation algorithm under Gap-ETH, even when parametrizing the runtime by $k$ [Dinur & Manurangsi, ITCS 2018]. In light of this, we systematically study several special cases of DSN and determine their parameterized approximability for the parameter $k$. For the bi-DSN$_\text{Planar}$ problem, the aim is to compute a planar optimum solution $N\subseteq G$ in a bidirected graph $G$, i.e., for every edge $uv$ of $G$ the reverse edge $vu$ exists and has the same weight. This problem is a generalization of several well-studied special cases. Our main result is that this problem admits a parameterized approximation scheme (PAS) for $k$. We also prove that our result is tight in the sense that (a) the runtime of our PAS cannot be significantly improved, and (b) it is unlikely that a PAS exists for any generalization of bi-DSN$_\text{Planar}$, unless FPT=W[1]. One important special case of DSN is the Strongly Connected Steiner Subgraph (SCSS) problem, for which the solution network $N\subseteq G$ needs to strongly connect a given set of $k$ terminals. It has been observed before that for SCSS a parameterized $2$-approximation exists when parameterized by $k$ [Chitnis et al., IPEC 2013]. We give a tight inapproximability result by showing that for $k$ no parameterized $(2-\varepsilon)$-approximation algorithm exists under Gap-ETH. Additionally we show that when restricting the input of SCSS to bidirected graphs, the problem remains NP-hard but becomes FPT for $k$

Hardness of Vertex Deletion and Project Scheduling

Assuming the Unique Games Conjecture, we show strong inapproximability results for two natural vertex deletion problems on directed graphs: for any integer $k\geq 2$ and arbitrary small $\epsilon > 0$, the Feedback Vertex Set problem and the DAG Vertex Deletion problem are inapproximable within a factor $k-\epsilon$ even on graphs where the vertices can be almost partitioned into $k$ solutions. This gives a more structured and therefore stronger UGC-based hardness result for the Feedback Vertex Set problem that is also simpler (albeit using the "It Ain't Over Till It's Over" theorem) than the previous hardness result. In comparison to the classical Feedback Vertex Set problem, the DAG Vertex Deletion problem has received little attention and, although we think it is a natural and interesting problem, the main motivation for our inapproximability result stems from its relationship with the classical Discrete Time-Cost Tradeoff Problem. More specifically, our results imply that the deadline version is NP-hard to approximate within any constant assuming the Unique Games Conjecture. This explains the difficulty in obtaining good approximation algorithms for that problem and further motivates previous alternative approaches such as bicriteria approximations.Comment: 18 pages, 1 figur

Inapproximability of H-Transversal/Packing

Given an undirected graph G=(V,E) and a fixed pattern graph H with k vertices, we consider the H-Transversal and H-Packing problems. The former asks to find the smallest subset S of vertices such that the subgraph induced by V - S does not have H as a subgraph, and the latter asks to find the maximum number of pairwise disjoint k-subsets S1, ..., Sm such that the subgraph induced by each Si has H as a subgraph. We prove that if H is 2-connected, H-Transversal and H-Packing are almost as hard to approximate as general k-Hypergraph Vertex Cover and k-Set Packing, so it is NP-hard to approximate them within a factor of Omega(k) and Omega(k / polylog(k)) respectively. We also show that there is a 1-connected H where H-Transversal admits an O(log k)-approximation algorithm, so that the connectivity requirement cannot be relaxed from 2 to 1. For a special case of H-Transversal where H is a (family of) cycles, we mention the implication of our result to the related Feedback Vertex Set problem, and give a different hardness proof for directed graphs