Question

I have a bipartite graph. I am looking for a maximum (1,n) “matching”, which means that each vertex from partitation A has n associated vertices from partition B.

The following figure shows a maximum (1,3) matching in a graph. Edges selected for the matching are red and unselected edges are black.

See figure http://www.freeimagehosting.net/uploads/9a8df2d97c.gif

This differs from the standard bipartite matching problem where each vertex is associate with only one other vertex, which could be called (1,1) matching with this notation.

If the matching cardinality (n) is not enforced but is an upper bound (vertices from A can have 0 < x <= n associated vertices from B), then the maximum matching can be found easily by transforming the graph to a flow network and finding the max flow. However, this does not guarantee that the maximum number of vertices from A will have n associated pairs from B.

Answer 1

This is NP-hard, reduction from maximum independent set problem. For any graph G you can construct (in polynomial time) an instance of your problem such that:

• Vertices in A represent vertices of G
• Each vertex of A is connected to exactly n vertices from B
• Any two vertices of A have a common neighbour in B if and only if they are connected in G. For this to be always possible, pick n=Δ(G).

Now the maximum ‘matching’ in the instance maps back to maximum independent set in G.