Lecture 13 : Int . Gap and Hardness of Priority Steiner Tree

The LP relaxation for the Priority Steiner Tree is similar to the relaxation used before for the Steiner Tree. Again, say a set S ⊆ V crosses the set R if S ∩ R 6= ∅ and R \S 6= ∅ there is at least one terminal in S and one not in S. For j ≤ k, let Sj be the collection of all sets which cross Rj . Finally, for a set S ⊆ V , let ∂S be the set of edges with one endpoint in S and one in V \S. Define E≤j to be ⋃j k=1 Ej , the set of edges of priority j or higher; the set of edges available to vertices of level j. The LP relaxation of Priority Steiner Tree then has a variable for each edge, representing the extent to which it is used by the fractional solution. The goal is to minimize the cost of edges used, while ensuring that every vertex in any Rj is connected to the root using only edges in E≤j ; this can be formulated in terms of crossing sets, giving the following LP relaxation: