We are given a graph G(V,E) with N nodes (Numbered from 0 to N-1) and exactly (N-1) two-way Edges.
Each edge in a graph has a positive cost C(u,v)(Edge weight).
The entire graph is such that there is a unique path between any pair of Nodes.
We are also given a List L of node number,at which bomb are placed.
Our aim is to damage/remove the edge from the graph such ,that after damaging/removing the edges from the graph ,there is no connection among the Bombs —
that is after damaging, there is no path between any two bombs.
The cost of damaging the Edge(u,v) = Edge weight(u,v).
So, we have to damage those edges, such that the total damaging cost is minimum.
Example:
Total Nodes=N=5
Total Bomb=Size of List L=3
List L={2,4,0}//Thats is at node number 2,4 and 0 bomb is placed...
Total Edges =N-1=4 edges are::
u v Edge-Weight
2 1 8
1 0 5
2 4 5
1 3 4
In this case the answer is ::
Total Damaging cost =(Edge Weight (2,4) + Edge Weight(0,1))
=5+5=10.
So when we remove the edge connecting node (2,4),
and the edge connecting node (0,1) ,there is no connection left
between any pair of machines in List {2,4,0};
Note any other,combinations of edges(that we damaged ) to achieve the
target goal ,needs more than 10 unit cost.
Constraints::
N(ie. Number of Nodes) <= 100,000
ListSize |L|(ie. Number of Bombs) <= N
1 <=Edge cost(u,v) <= 1000,000
What i had done?
Until now, I had not found any efficient way 🙁 .
Further, as the number of nodes is N, the number of edges is exactly N-1 and the entire graph is such there is a Unique path between any pair of Nodes, I got a conclusion that the graph is a TREE.
I tried to modify the Kruskal algorithm but that didn’t help me either.
Thanks!
This is the Multiway Cut problem in trees. It can be solved in polynomial time by a straightforward dynamic programming. See Chopra and Rao: “On the multiway cut polyhedron“, Networks 21(1):51–89, 1991.