Directed Steiner Approximation Algorithms

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Directed Steiner Approximation Algorithms
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Steiner Tree Problem in Graphs

• Given Weighted graph G(V,E,cost) and terminals
  S ? V
• Find Minimum-cost tree T within G spanning S
• Complexity Max SNP-hard Bern Plassmann, 1989
• even in complete graphs with
  edge costs 1 2
• Geometric STP NP-hard Garey
  Johnson, 1977
• but has PTAS Arora, 1996

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Undirected and Directed versions of Steiner tree
problem
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Approximation Ratios in Graphs
(New performance ratio)
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Idea of Approximation Algorithms

• One destination Shortest path is the optimal
  solution.
• Two destinations There is one start point where
  the path from the root splits into two paths to
  the destinations. As there are only n nodes on
  which the path can be located, the best position
  can be found in O(n) time after the shortest
  paths for the terminals have been calculated.
• Three destinations The number of start point is
  two. The optimal location of these points can be
  found in O(nm) time. The factor m comes from the
  time required to calculate the shortest path from
  the first start point to all possible locations
  of the second start point.
• gets beyond there the time required for an
  optimal solution grows out of hand very fast.

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Idea of Roos Algorithm
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Pseudo code of Roos Algorithm
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Pseudo code of Charikar's Algorithm
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Idea of Charikar's Algorithm
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Performance Ratio of Charikar's Algorithm
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Idea of Zelikovskys Algorithm
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Pseudo code of TM Algorithm
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Theorem of TM Algorithm
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Pseudo code of KMB Algorithm
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Theorem of KMB Algorithm