Sina Dehghani, Soheil Ehsani, Saeed Seddighhin Stein iner er tree: - - PowerPoint PPT Presentation

Sina Dehghani, Soheil Ehsani, Saeed Seddighhin

 Stein

iner er tree: Given an edge-weighted graph 𝐻 = 〈 𝑊, 𝐹, 𝑥〉 and a subset 𝑇 ⊂ 𝑊 of required vertices. A Steiner tree is a tree in 𝐻 that spans all vertices of 𝑇.

 Stein

iner er fores est: Given an edge-weighted graph 𝐻 = 〈 𝑊, 𝐹, 𝑥〉 and a list of pairs 𝑣1, 𝑤𝑗 , 𝑣2, 𝑤2 , … , (𝑣𝑙, 𝑤𝑙) of vertices. A Steiner forest is a subgraph of 𝐻 in which every 𝑤𝑗 is reachable from 𝑣𝑗.

 Th

There ere are e ma many y va varian ants ts of St Steiner einer net etwork work problem blem.

• Points

ints in the plane. ane.

• Weight

ights s on the ve vertices rtices.

• Dir

irected ected Stei einer ner networ work. k.

• Pric

ice collectin llecting g St Steiner einer network twork problems

• blems.
• Online

line St Steiner einer netwo work rk problems.

• blems.
• Degr

gree ee-bo bounde nded d St Steiner iner network twork problems

• blems.

• You have the whole network at the beginning.
• Demand vertices/pairs come one by one.

 In the Steiner tree problem once a demand vertex is added to set 𝑇, you have to add some vertices and edges to the subgraph such that the it connects the new node to other demand vertices.  In the Steiner forest problem once a pair of vertices is added to the list, you have to add some vertices and edges to the subgraph such that the newly added vertices become connected in the subgraph.

This is addi ditional tional constrai nstraint nt makes the problem

• blem hard

rder, er, eve ven n for

• r the case where

ere we want t to find ind a subtree tree that t spans ns all l of the ve vert rtic ices. es.

• The original problem is equivalent to finding an MST of the graph

which can be solved in polynomial time.

• If you bound the degree of vertices by 2, it becomes equivalent to

finding the shortest Hamiltonian path of the graph which is indeed NP-hard.