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 𝑣 1 , 𝑤 𝑗 , 𝑣 2 , 𝑤 2 , … , (𝑣 𝑙 , 𝑤 𝑙 ) of vertices. A Steiner forest is a subgraph of 𝐻 in which every 𝑤 𝑗 is reachable from 𝑣 𝑗 .
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 oblems. ◦ Online line St Steiner einer netwo work rk problems. oblems. ◦ Degr gree ee-bo bounde nded d St Steiner iner network twork problems oblems.
◦ 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 oblem hard rder, er, eve ven n for or 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.
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