Approximating Rooted Steiner Networks

Abstract

The Directed Steiner Tree (DST) problem is a cornerstone problem in network design. We focus on the generalization of the problem with higher connectivity requirements. The problem with one root and two sinks is APX-hard. The problem with one root and many sinks is as hard to approximate as the directed Steiner forest problem, and the latter is well known to be as hard to approximate as the label cover problem. Utilizing previous techniques, we strengthen these results and extend them to undirected graphs. Specifically, we give an Ω( k ϵ ) hardness bound for the rooted k -connectivity problem in undirected graphs. As a consequence, we obtain an Ω( k ϵ ) hardness bound for the undirected subset k -connectivity problem. Additionally, we give a result on the integrality ratio of the natural linear programming relaxation of the directed rooted k -connectivity problem.

Parameters of Two-Prover-One-Round Game and The Hardness of Connectivity Problems

“…In this problem, we are given an undirected graph G = (V, E), a root vertex r and a set of terminals T ; the goal is to find a minimum-cost subgraph that has k openly (vertex) disjoint paths from the root vertex r to each terminal t ∈ T . For arbitrary k, the best known approximation ratio of this problem is O(k log k) by Nutov [24], and it was shown by Cheriyan, Laekhanukit, Naves and Vetta [7] that the dependence on k cannot be taken out because the problem does not admit o(k σ )-approximation, for some (very) small constant σ > 0, unless P = NP. However, when k is larger than the number of demands (or terminals) D, a trivial D-approximation algorithm does exist and yields a better approximation ratio than the O(k log k)-approximation algorithm.…”

An Improved Approximation Algorithm for the Minimum Cost Subset k-Connected Subgraph Problem

“…The hardness result in Theorem 2.4 together with the hardness of the rooted subset k-connectivity problem by Cheriyan, Laekhanukit, Naves and Vetta [4] implies the hardness of Ω(k ), for the subset k-connectivity problem, where > 0 is some fixed constant.…”

“…[14] no non-trivial approximation algorithm was known for the general case of the subset-k-connectivity problem until the work of Chakraborty, Chuzhoy and Khanna [3]. They presented an O(k O(k 2 ) • log 4 |T |)-approximation algorithm for the rooted version of our problem, namely the rooted subset k-connectivity problem. There, given a root vertex r and a set of terminals T , the goal is to find a minimum cost subgraph that has k openly disjoint paths from the root vertex r to every terminal in T .…”

“…There, given a root vertex r and a set of terminals T , the goal is to find a minimum cost subgraph that has k openly disjoint paths from the root vertex r to every terminal in T . Chakraborty et al showed how to solve the subset k-connectivity problem by applying the rooted subset k-connectivity algorithm k times, thus obtaining an O(k O(k 2 ) •log 4 |T |)-approximation algorithm. Recently, in a series of developments [3,7,8,21,18], the approximation guarantees for the rooted subset k-connectivity problem has been steadily improved.…”

Fault-Tolerant Edge-Disjoint Paths -- Beyond Uniform Faults

“…, t k }. According to [7], even the complexity of 1-2-connected Directed 2 Steiner Tree is open, which is the following variant of 2-connected Directed k Steiner Tree: we have only two terminals t 1 and t 2 and aim to find two disjoint s-t 1 paths and one s-t 2 path of minimal total cost. Note that 2-connected Directed k Steiner Tree is a generalization of Directed Steiner Tree and therefore does not admit a polynomial-time log 2−ε n approximation algorithm unless NP ⊆ ZTIME(n polylog(n) ) [14].…”