An O (kn)-Time Algorithm to Solve Steiner (k , k ′)-Eccentricity on Trees.

Steiner (k , k ′) -eccentricity on a given fixed k ′ -subset in a graph G is the maximum Steiner distance over all k-subsets of V (G) which contain the fixed k ′ -set, where the Steiner distance of a set is the size of a minimum Steiner tree on this set in a graph. Let R ⊆ V (T) be the given fixed k ′ -subset in a tree T. Let k 1 and k 2 be two integers such that k 1 ≥ k 2 ≥ k ′ . We prove that, in a tree, every optimal solution of Steiner (k 1 , k ′) -eccentricity on R takes some optimal solution of Steiner (k 2 , k ′) -eccentricity on R as a partial solution. On the other hand, every optimal solution of Steiner (k 2 , k ′) -eccentricity on R is part of some optimal solution of Steiner (k 1 , k ′) -eccentricity on the set R in a tree. Finally, we present an O (k n) -time algorithm to solve Steiner (k , k ′) -eccentricity on a given fixed k ′ -set in trees. [ABSTRACT FROM AUTHOR]

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