CODEWITHSUNDEEP
algorithmtime-complexitygraph-theorynp-completebipartite
Lingxi
Lingxi

Reputation: 14987

Is this bipartite graph optimization task NP-complete?

I have been trying to find out a polynomial-time algorithm to solve this problem, but in vain. I'm not familiar with the NP-complete thing. Just wondering whether this problem is actually NP-complete, and I should not waste any further effort trying to come up with a polynomial-time algorithm.

The problem is easy to describe and understand. Given a bipartite graph, what is the minimum number of vertices you have to select from one vertex set, say A, so that each vertex in B is adjacent to at least one selected vertex.

Upvotes: 1

Views: 2748

Answers (1)

j_random_hacker
j_random_hacker

Reputation: 51296

Unfortunately, this is NP-hard; there's an easy reduction from Set Cover (in fact it's arguably just a different way of expressing the same problem). In Set Cover we're given a ground set F, a collection C of subsets of F, and a number k, and we want to know if we can cover all n ground set elements of F by choosing at most k of the sets in C. To reduce this to your problem: Make a vertex in B for each ground element, and a vertex in A for each set in C, and add an edge uv whenever ground element v is in set u. If there was some algorithm to efficiently solve the problem you describe, it could solve the instance I just described, which would immediately give a solution to the original Set Cover problem (which is known to be NP-hard).

Interestingly, if we are allowed to choose vertices from the entire graph (rather than just from A), the problem is solvable in polynomial time using bipartite maximum matching algorithms, due to Kőnig's Theorem.

Upvotes: 3

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