Approximating Rooted Steiner Networks

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

“…(In fact, the k 1/2− -hardness of this problem can be derived from combining the result in [7] and [13].) For the case of undirected graphs, the hardness are k 1/10− and D 1/4− , for any constant > 0, and this also gives the same bound for the hardness of the subset k-connectivity problem.…”

“…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.…”

“…These problems have hardness of the form k σ , where σ is a small constant that has not been calculated. (See [24,7,3,11]). The common source of hardness of these problems is the label cover problem (a.k.a., 2P1R) with parallel repetition.…”

“…In the rooted k-connectivity problem, we are given a directed or undirected graph G = (V, E) on n vertices with cost c e on each edge e ∈ E, a root vertex r, a set of terminals T ⊆ V − {r} and a connectivity requirement k. The goal is to find a minimum-cost subgraph G = (V, E ) of G such that G has k openly (vertex) disjoint paths from r to each terminal t ∈ T . This problem has been studied intensively in [3,9,5,10,24,25,6,7]. The rooted k-connectivity problem is a fundamental network design problem with vertex-connectivity requirements, and it lies at the bottom of the complexity hierarchy of the vertexconnectivity problems.…”

“…Non-trivial approximation algorithms for the rooted k-connectivity problem are known only for the undirected case, and the best known approximation ratio is O(k log k) by Nutov [24]; however, the approximation ratio surpasses that of the trivial algorithm only when k > |T |. On the negative side, Cheriyan, Laekhanukit, Naves and Vetta [7] recently showed that the rooted kconnectivity problem on both directed and undirected graphs are hard to approximate to within a factor of k σ for some fixed σ > 0 (the constants σ are different in directed and undirected cases). However, the constants σ obtained are small and have not been explicitly calculated.…”