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Explore a general result on greedy algorithms for k-restricted Steiner trees with a non-integer potential function. The theorem shows the approximation guarantee, Loss(T) operation, and the proof outline for optimal solutions.
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Chapter 3 Restriction (2) Greedy k-restricted Steiner tree Ding-Zhu Du
A general result on greedy algorithm With non-integer potential function Consider a monotone increasing, submodular function Consider the following problem: where is a nonnegative cost function
Theorem Suppose in Greedy Algorithm G, selected x always satisfies Then its p.r. where
Proof. Let be obtained by Greedy Algorithm G. Denote be an optimal solution. Let Denote
Note that There exists i such that
Let Let Note that So
Note Hence,
Consider where is the length of MST on P after terminals in each connected component of H are contracted into a point. Consider the set of all full component of size at most k. Theorem. is a monotone increasing submodular function on
consider each For k >2, as a set of edges in a spanning tree on terminals. For
iff iff i.e.,
For k=2, is the length of a longest edge in the path connecting two endpoints of , in MST(A). x
x x x x y
x x x y
Greedy Algorithm G is Theorem -approximation for . Greedy Algorithm G produces approximation solution for SMT with length at most
Lemma Proof.
Lemma Proof
