Subgraph Isomorphism checking in Multigraphs
Submultigraph isomorphism is clearly NP complete:


it's easily seen to be in NP (an isomorphism can be specified in space polynomial in the size of the graphs, and it's easy to check that a proposed isomorphism is valid);
it includes the NP-complete subgraph isomorphism problem as a special case.


The more interesting question is what the best known algorithms are; I'm afraid I can't help with that.