Max-density Subgraph with Prescribed Node Problem : given a Hypergraph H(v; E) And

a prescribed node p, nd the maximum density subgraph which contains node p. This can be solved using the 0 ? 1 fractional programming technique in 24] by assigning the variable corresponding to p to 1, and transforming the resulting fractional expression to a series of ow computations. Max-Density Subgraph with Excluded Node problem : Given a hypergraph H(V; E) and a prescribed node p, nd the maximum density subgraph which does not contain node p. This problem can be solved in the same time complexity as the MDS problem. We can remove node p and all edges incident from p, and solve the MDS problem in the remaining graph. It is reasonable to suspect that the Prescribed and Excluded Node variants are useful for certain partitioning and clustering purposes. 7 Acknowledgments We are very grateful to Lars Hagen for his valuable comments, and Laura Sanchis for providing k-way FM code 26]. Also thanks to L. T. Liu for valuable discussion.