On the clustered Steiner tree problem

Abstract

We investigate the Clustered Steiner tree problem on metric graphs, which is a variant of the Steiner minimum tree problem. In this problem, the required vertices are partitioned into clusters, and the subtrees spanning different clusters must be disjoint in a feasible clustered Steiner tree. In this paper, it is shown that the problem is NPhard even if the inter-cluster tree and all the local topologies are given, where a local topology specifies the tree structure of required vertices in the same cluster. We show that the Steiner ratio of this problem is lower and upper bounded by three and four, respectively. We also propose a (ρ + 2)-approximation algorithm, where ρ is the approximation ratio for the Steiner minimum tree problem, and the approximation ratio can be improved to ρ + 1 if the local topologies are given. Two variants of this problem are also studied. When the goal is to minimize the inter-cluster cost and ignore the cost of local trees, the problem can be solved in polynomial time. But it is NP-hard if we ask for the minimum cost of local trees among all solutions with minimum inter-cluster cost.

The clustered selected-internal Steiner tree problem

“…Although the SISTP is defined by a group of vertices (terminals), some applications of computer and transportation network routing are considered in more than one group of vertices [1,9,11,16,23,30]. Chisman [9] presented a variant of the traveling salesman problem [10], called as the clustered traveling salesman problem.…”

“…Chisman [9] presented a variant of the traveling salesman problem [10], called as the clustered traveling salesman problem. Wu and Lin [30] proposed another variant of the Steiner tree problem, called as the clustered Steiner tree problem. Given a complete graph G = (V, E), with a nonnegetive cost function on edges, a subset R ⊂ V, and a partition…”

“…The clustered Steiner tree problem (CSTP for short) is concerned with the determination of a clustered Steiner tree T for R in G with minimum cost. Wu and Lin [30] showed that the CSTP is NP-complete and proposed a (ρ + 2)-approximation algorithm for the CSTP on metric graphs. In this paper, we presented a variant of the SISTP and the CSTP, called as the clustered selected-internal Steiner tree problem.…”

“…Then, for some communication networks, sometimes the edges are divided into two levels: inter-cluster or intra-cluster, possibly with different costs, qualities, and capacities. After the multicast tree is constructed, the communications between hosts in the same cluster should be routed locally rather than globally for the sake of capacity consideration or the simpleness of routing protocols [30]. If all local topologies are given [30].…”

“…After the multicast tree is constructed, the communications between hosts in the same cluster should be routed locally rather than globally for the sake of capacity consideration or the simpleness of routing protocols [30]. If all local topologies are given [30]. The purpose is to design the inter-cluster topology, as well as the possible insertion of local Steiner vertices without violating their topologies.…”