On the Complexities of the Incremental Bottleneck and Bottleneck Terminal Steiner Tree Problems

Given a graph G = (V, E) with non-negative edge lengths, a subset R ⊂ V, a Steiner tree for R in G is an acyclic subgraph of G interconnecting all vertices in R and a terminal Steiner tree is defined to be a Steiner tree in G with all the vertices of R as its leaves. A bottleneck edge of a Steiner tree is an edge with the largest length in the Steiner tree. The bottleneck Steiner tree problem (BSTP) (respectively, the bottleneck terminal Steiner tree problem (BTSTP)) is to find a Steiner tree (respectively, a terminal Steiner tree) for R in G with minimum length of a bottleneck edge. For any arbitrary tree T, len_b(T) denotes the length of a bottleneck edge in T. Let T_opt(G, BSTP) and T_opt(G, BTSTP) denote the optimal solutions for the BSTP and the BTSTP in G, respectively. Given a graph G = (V, E) with non-negative edge lengths, a subset E₀ ⊂ E, a number h = |E \E₀|, and a subset R ⊂ V, the incremental bottleneck Steiner tree problem (respectively, the incremental bottleneck terminal Steiner tree problem) is to find a sequence of edge sets {E₀ ⊂ E₁ ⊂ E₂ ⊂ ... ⊂ E_h = E} with |E_i\E_i-1| = 1 such that Σ^h_i=1 len_b(T_opt(G_i, BSTP)) (respectively, Σ^h_i=1 len_b(T_opt(G_i, BTSTP))) is minimized, where G_i = (V, E_i). In this paper, we prove that the incremental bottleneck Steiner tree problem is NP-hard. Then we show that there is no polynomial time approximation algorithm achieving a performance ratio of (1-ε) × ln |R|, 0 <; ε <; 1, for the incremental bottleneck terminal Steiner tree problem unless NP ⊆ DTIME(|R|^{log log |R|}).