The bottleneck selected‐internal and partial terminal Steiner tree problems

Abstract

Given a complete graph G = (V , E ), a positive length function on edges, and two subsets R of V and R of R, the selected-internal Steiner tree is defined to be an acyclic subgraph of G spanning all vertices in R such that no vertex in R is a leaf of the subgraph. The bottleneck selected-internal Steiner tree problem is to find a selected-internal Steiner tree T for R and R in G such that the length of the largest edge in T is minimized. The partial terminal Steiner tree is defined to be an acyclic subgraph of G spanning all vertices in R such that each vertex in R is a leaf of the subgraph. The bottleneck partial terminal Steiner tree problem is to find a partial terminal Steiner tree T for R and R in G such that the length of the largest edge in T is minimized. In this article, we show that the bottleneck selected-internal Steiner tree problem is NP-complete. We also show that if there is a (2− )-approximation algorithm, > 0, for the bottleneck selected-internal Steiner tree problem on metric graphs (i.e., a complete graph and the lengths of edges satisfy the triangle inequality), then P=NP. Then we extend to show that if there is an (α(|V |) − )-approximation algorithm, > 0, for the bottleneck selected-internal Steiner tree problem, then P=NP, where α(|V |) is any computable function of |V |. Moreover, we present an approximation algorithm with performance ratio of 3 for the bottleneck selected-internal Steiner tree problem on metric graphs. Finally, we present an exact algorithm of O(|V | 2 log |V |) time for the bottleneck partial terminal Steiner tree problem.

The clustered selected-internal Steiner tree problem

“…We let A h (G, u, v) be the algorithm for finding a Hamiltonian path between the two vertices u and v in the G 3 by Karaganis' proof [22], whose time-complexity is O(|V| 2 ). See Appendixm or Chen [7] for more details of this algorithm. For a tree T = (V T , E T ) and (u, w), (w, v) ∈ E T , we define the shortcut between u and v is to replace edges (u, w) and (w, v) with (u, v).…”

“…Appendix [7] Karaganis [22] proved that the cube of a connected graph G with at least three vertices is Hamiltonian-connected, i.e., there exists a Hamiltonian path between any two vertices. In this proof, Karaganis let T be a spanning tree of G and construct a Hamiltonian path between any two vertices v a and v b in T 3 recursively, in which T 3 is the cube of T .…”