Proof of NP-completeness via reduction

[goraemon]

Homework Statement

You are given some undirected graph G = (V, E), along with a set S which consists of 0 or more pairs of G's edges.

As an example, a complete graph on 3 vertices (a triangle, basically) would be described as follows: G = (V, E) = ({v1, v2, v3}, {v1v2, v2v3, v1v3}). A set S could consist of 0 or more pairs of G's edges. For example, a set S consisting of a single element could be simply {(v1v2, v2v3)}.

Now, we'll define M as a subset of edges that has at most one edge incident to each vertex in G, and at most one edge from each pair in S.

Given a graph G, some subset of edge-pairs S, and some integer k >= 1, show that the problem of determining whether G contains a subset M of size at least k is NP-complete.

Homework Equations

Use reduction of a known NP-complete problem.

The Attempt at a Solution

I'm quite lost as to where to start, and even which known NP-complete problem to use for the reduction. Since it's a graph problem, I'm thinking perhaps something like vertex cover, independent set, clique, etc. might be used for reduction, but have no idea which or how. Thanks for any assistance.

 

[KataKoniK]

Thank you for posting your question. This problem can be reduced to the well-known NP-complete problem of Maximum Matching in a graph.

First, let's define the decision problem for Maximum Matching (MM): Given an undirected graph G = (V, E) and an integer k >= 1, does G have a matching of size at least k?

Now, we can show that the problem described in the forum post can be reduced to MM.

Given an instance of the problem described in the forum post, we can create a new graph G' = (V', E') as follows:
1. For each edge in G, add two new vertices to V', one for each endpoint of the edge.
2. For each pair of edges in S, add a new vertex to V'.
3. For each edge in G and each pair of edges in S, add an edge between the two corresponding vertices in V'.

Now, we can see that G' has a matching of size at least k if and only if G has a subset M of size at least k that satisfies the given conditions.

Therefore, we have reduced the problem described in the forum post to MM, which is known to be NP-complete. This proves that the problem described in the forum post is also NP-complete.

I hope this helps. Let me know if you have any further questions.