Approximation Algorithms for Non-single-minded Profit-Maximization Problems with Limited Supply

We consider profit-maximization problems for combinatorial auctions with non-single minded valuation functions and limited supply. We obtain fairly general results that relate the approximability of the profit-maximization problem to that of the corresponding social-welfare-maximization (SWM) problem, which is the problem of finding an allocation (S1, . . . , Sn) satisfying the capacity constraints that has maximum total value ∑ j vj(Sj). Our results apply to both structured valuation classes, such as subadditive valuations, as well as arbitrary valuations. For subadditive valuations (and hence submodular, XOS valuations), we obtain a solution with profit OPTSWM /O(log cmax), where OPTSWM is the optimum social welfare and cmax is the maximum item-supply; thus, this yields anO(log cmax)-approximation for the profit-maximization problem. Furthermore, given any class of valuation functions, if the SWM problem for this valuation class has an LP-relaxation (of a certain form) and an algorithm “verifying” an integrality gap of α for this LP, then we obtain a solution with profit OPTSWM /O(α log cmax), thus obtaining an O(α log cmax)approximation. The latter result implies an O( √ m log cmax)-approximation for the profit maximization problem for combinatorial auctions with arbitrary valuations, and an O(log cmax)-approximation for the non-single-minded tollbooth problem on trees. For the special case, when the tree is a path, we also obtain an incomparableO(logm)-approximation (via a different approach) for subadditive valuations, and arbitrary valuations with unlimited supply. 4