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Table 1 Approximation bounds for the various Steiner Network Problems in their classic setting and condition setting

From: Connectivity problems on heterogeneous graphs

Problems Classic Condition
Lower bound Upper bound Lower bound(s) Upper bound(s)
Steiner Forest 1.01 [19] 2 [18] \(C-\epsilon , k-\epsilon \) 2Ck
Directed Steiner Network \(k^{1/4 - o(1)}\) [27] \(k^{1/2+\epsilon }\) [21, 22] \(C-\epsilon , k-\epsilon \) \(C \cdot k^{1/2+\epsilon }, k\)
Undirected Shortest Path N/A 1 \(C-\epsilon \) C
Directed Shortest Path N/A 1 \(C-\epsilon \) C
Steiner Tree 1.01 [19] 1.39 [17] \(C-\epsilon \) 1.39C
Prize-Collecting Steiner Tree 1.01 [19] 1.97 [26] \(C-\epsilon \) 1.97C
  1. For the classic problems, we have indicated the papers in which the bounds are shown. For the condition problems, all the lower bounds are developed in the present work; all the upper bounds are the naive bounds obtained from the “union of shortest paths” heuristic, or from applying the best known approximation algorithm for the appropriate classic Steiner problem to each condition, then taking the union of those solutions