WINE 2010: Internet and Network Economics pp 462-472

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

• Khaled Elbassioni
• Mahmoud Fouz
• Chaitanya Swamy
Conference paper

DOI: 10.1007/978-3-642-17572-5_39

Part of the Lecture Notes in Computer Science book series (LNCS, volume 6484)
Cite this paper as:
Elbassioni K., Fouz M., Swamy C. (2010) Approximation Algorithms for Non-single-minded Profit-Maximization Problems with Limited Supply. In: Saberi A. (eds) Internet and Network Economics. WINE 2010. Lecture Notes in Computer Science, vol 6484. Springer, Berlin, Heidelberg

## Abstract

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 ∑ jvj(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 Open image in new window, where Open image in new window is the optimum social welfare and c max is the maximum item-supply; thus, this yields an O(logc max )-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 Open image in new window, thus obtaining an O(α\log c_{\max})-approximation. The latter result implies an $$O(\sqrt m\log c_{\max})$$-approximation for the profit maximization problem for combinatorial auctions with arbitrary valuations, and an O(logc max )-approximation for the non-single-minded tollbooth problem on trees. For the special case, when the tree is a path, we also obtain an incomparable O(logm)-approximation (via a different approach) for subadditive valuations, and arbitrary valuations with unlimited supply.

## Authors and Affiliations

• Khaled Elbassioni
• 1
• Mahmoud Fouz
• 2
• Chaitanya Swamy
• 3
1. 1.Max-Planck-Institut für InformatikSaarbrückenGermany
2. 2.FR InformatikUniversität des SaarlandesSaarbrückenGermany
3. 3.Dept. of Combinatorics and OptimizationUniv. WaterlooWaterloo