Directed Steiner Network Research Articles

Parameterized Approximation Algorithms for Bidirected Steiner Network Problems

The D irected S teiner N etwork (DSN) problem takes as input a directed graph G =( V , E ) with non-negative edge-weights and a set D ⊆ V × V of k demand pairs. The aim is to compute the cheapest network N⊆ G for which there is an s\rightarrow t path for each ( s , t )∈ 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 -DSNP lanar problem, the aim is to compute a solution N⊆ G whose cost is at most that of an optimum planar solution 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) no PAS exists for any generalization of bi-DSNP lanar , under standard complexity assumptions. The techniques we use also imply a polynomial-sized approximate kernelization scheme (PSAKS). Additionally, we study several generalizations of bi -DSNP lanar and obtain upper and lower bounds on obtainable runtimes parameterized by k . One important special case of DSN is the S trongly C onnected S teiner S ubgraph (SCSS) problem, for which the solution network N⊆ G needs to strongly connect a given set of k terminals. It has been observed before that for SCSS a parameterized 2-approximation exists for parameter k [Chitnis et al., IPEC 2013]. We give a tight inapproximability result by showing that for k no parameterized (2 − ε)-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 .

Parameterized Complexity of Directed Steiner Network with Respect to Shared Vertices and Arcs

We consider the directed Steiner network problem, where given a weighted directed graph [Formula: see text] and [Formula: see text] pairs of vertices [Formula: see text], one has to find the minimum weight subgraph [Formula: see text] of [Formula: see text] that contains path from [Formula: see text] to [Formula: see text] for all [Formula: see text]. We search for the main cause of the complexity of the problem and introduce the notions of junction vertex and shared segment in an optimal solution. We study the parameterized complexity of the problem with respect to these parameters and achieve several parameterized hardness results. Also, we present two output-sensitive algorithms for the problem, which are much simpler and easier to implement than the previously best known one; and in addition, when the paths corresponding to the input pairs have few shared vertices or arcs in the optimal solution, these algorithms outperform the previous one.

Set connectivity problems in undirected graphs and the directed steiner network problem

In the generalized connectivity problem, we are given an edge-weighted graph G = ( V , E ) and a collection D = {( S 1 , T 1 ), …, ( S k , T k )} of distinct demands each demand ( S i , T i ) is a pair of disjoint vertex subsets. We say that a subgraph F of G connects a demand ( S i , T i ) when it contains a path with one endpoint in S i and the other in T i . The goal is to identify a minimum weight subgraph that connects all demands in D . Alon et al. (SODA '04) introduced this problem to study online network formation settings and showed that it captures some well-studied problems such as Steiner forest, facility location with nonmetric costs, tree multicast, and group Steiner tree. Obtaining a nontrivial approximation ratio for generalized connectivity was left as an open problem. We describe the first poly-logarithmic approximation algorithm for generalized connectivity that has a performance guarantee of O (log 2 n log 2 k ). Here, n is the number of vertices in G and k is the number of demands. We also prove that the cut-covering relaxation of this problem has an O (log 3 n log 2 k ) integrality gap. Building upon the results for generalized connectivity, we obtain improved approximation algorithms for two problems that contain generalized connectivity as a special case. For the directed Steiner network problem, we obtain an O ( k 1/2 + ϵ ) approximation which improves on the currently best performance guarantee of Õ ( k 2/3 ) due to Charikar et al. (SODA '98). For the set connector problem, recently introduced by Fukunaga and Nagamochi (IPCO '07), we present a poly-logarithmic approximation; this result improves on the previously known ratio which can be Ω( n ) in the worst case.

Parameterized Complexity of Arc-Weighted Directed Steiner Problems

We start a systematic parameterized computational complexity study of three NP-hard network design problems on arc-weighted directed graphs: directed Steiner tree, strongly connected Steiner subgraph, and directed Steiner network. We investigate their parameterized complexities with respect to the three parameterizations: “number of terminals,” “an upper bound on the size of the connecting network,” and the combination of these two. We achieve several parameterized hardness results as well as some fixed-parameter tractability results, in this way extending previous results of Feldman and Ruhl [SIAM J. Comput., 36 (2006), pp. 543–561].

The Directed Steiner Network Problem is Tractable for a Constant Number of Terminals

We consider the Directed Steiner Network problem, also called the Point‐to‐Point Connection problem. Given a directed graph G and p pairs $\{ (s_1,t_1), \dotsc, (s_p,t_p) \}$ of nodes in the graph, one has to find the smallest subgraph H of G that contains paths from $s_i$ to $t_i$ for all i. The problem is NP‐hard for general p, since the Directed Steiner Tree problem is a special case. Until now, the complexity was unknown for constant $p \geq 3$. We prove that the problem is polynomially solvable if p is any constant number, even if nodes and edges in G are weighted and the goal is to minimize the total weight of the subgraph H. In addition, we give an efficient algorithm for the Strongly Connected Steiner Subgraph problem for any constant p, where given a directed graph and p nodes in the graph, one has to compute the smallest strongly connected subgraph containing the p nodes.