Reducing induced width of set membership graph

I have a number of sets, and a related graph where set elements are connected if they are in a set with each other. I need to find an ordering for the set elements which will result in a low induced width for the graph (I understand that finding the ordering that yields the minimal induced开发者_C百科 width is np-hard).

1. What is the best heuristic for ordering a graph to get a low induced width?

2. Is there a way to find a good ordering without generating the actual graph and working directly from the sets?

I'm doing this because I'm trying to implement an algorithm which is exponential on the induced width of the ordering chosen.

Edit: some definitions.

Ordered graph - the graph mentioned above, with a total ordering applied to the nodes.

Parent node - For a given node, an adjacent node which precedes the node in the ordering is a parent of that node.

Induced graph - The graph obtained by adding an edge between any two nodes that are parents of the same node.

Width of an ordered graph - The greatest number of parents any node in the ordered graph has.

Induced width - Width of the induced graph.