A decision problem: For a given graph G and numbers 'a','b' it is required to be answered whether there is a set of 'a' vertices which have a cumulative neighborhood of size at least 'b'. How do we show that this problem is NPC?
Asked
Active
Viewed 122 times
1 Answers
1
I think that if you could solve this in polynomial time you could solve https://en.wikipedia.org/wiki/Maximum_cut in polynomial time. According to the article the decision problem in Max-Cut is "Given a graph G and an integer k, determine whether there is a cut of size at least k in G".
Given something that solves your a/b version of the problem I would solve the Max-Cut version by setting b=k and trying a=1,2,3..size of graph, which would still have cost polynomial in the input size. (or possibly b=k+a depending on the details of what you mean by neighborhood size).
(So yes, I think your problem is NP-complete).
mcdowella
- 19,301
- 2
- 19
- 25
-
I kind of more found it reasonable to argue that it is NPC by reducing the Knapsack Problem to this one. Only thing is that nodes can have common neighbors and that's where the knapsack problem needs to be tweaked to reduce it to the problem in the question. ON a different note , the KP can also be reduced to maxcut decision variant. – CoderAmateur Apr 23 '17 at 13:29