  We study the problem of characterizing revenue optimal auctions for
single-minded buyers. Each buyer is interested only in a specific bundle of
items and has a value for the same. Both his bundle and its value are his
private information. The bundles that buyers are interested in and their
corresponding values are assumed to be realized from known probability
distributions independent across the buyers. We identify revenue optimal
auctions with a simple structure, if the conditional distribution of any
buyer's valuation is nondecreasing, in the hazard rates ordering of probability
distributions, as a function of the bundle the buyer is interested in. The
revenue optimal auction is given by the solution of a maximum weight
independent set problem. We provide a novel graphical construction of the
weights and highlight important properties of the resulting auction.
