  For a graph formed by vertices and weighted edges, a generalized minimum
dominating set (MDS) is a vertex set of smallest cardinality such that the
summed weight of edges from each outside vertex to vertices in this set is
equal to or larger than certain threshold value. This generalized MDS problem
reduces to the conventional MDS problem in the limiting case of all the edge
weights being equal to the threshold value. We treat the generalized MDS
problem in the present paper by a replica-symmetric spin glass theory and
derive a set of belief-propagation equations. As a practical application we
consider the problem of extracting a set of sentences that best summarize a
given input text document. We carry out a preliminary test of the statistical
physics-inspired method to this automatic text summarization problem.
