Dense Steiner problems: Approximation algorithms and inapproximability
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The Steiner Tree problem is a classical problem in combinatorial optimization: the goal is to connect a set T of terminals in a graph G by a tree of minimum size. Karpinski and Zelikovsky (1996) studied the δ-dense version of Steiner Tree, where each terminal has at least δ |V(G)∖ T| neighbours outside T, for a fixed δ > 0. They gave a PTAS for this problem. We study a generalization of pairwise δ-dense Steiner Forest, which asks for a minimum-size forest in G in which the nodes in each terminal set T_1,...,T_k are connected, and every terminal in T_i has at least δ |T_j| neighbours in T_j, and at least δ|S| nodes in S = V(G)∖ (T_1∪...∪ T_k), for each i, j in {1,..., k} with i≠ j. Our first result is a polynomial-time approximation scheme for all δ > 1/2. Then, we show a (13/12+ε)-approximation algorithm for δ = 1/2 and any ε > 0. We also consider the δ-dense Group Steiner Tree problem as defined by Hauptmann and show that the problem is APX-hard.
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