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Abstract

Consider the following problem. A seller has infinite copies of n products represented by nodes in a graph. There are m consumers, each has a budget and wants to buy two products. Consumers are represented by weighted edges. Given the prices of products, each consumer will buy both products she wants, at the given price, if she can afford to. Our objective is to help the seller price the products to maximize her profit.This problem is called graph vertex pricing (GVP) problem and has resisted several recent attempts despite its current simple solution. This motivates the study of this problem on special classes of graphs. In this paper, we study this problem on a large class of graphs such as graphs with bounded treewidth, bounded genus and k-partite graphs.We show that there exists an FPTAS for GVP on graphs with bounded treewidth. This result is also extended to an FPTAS for the more general single-minded pricing problem. On bounded genus graphs we present a PTAS and show that GVP is NP-hard even on planar graphs. * Work partially done while at the Department of Computer Science, University of Chicago, Chicago, IL. † Supported in part by the following research grants: Nanyang Technological University grant M58110000, Singapore Ministry of Education (MOE) Academic Research Fund (AcRF) Tier 2 grant MOE2010-T2-2-082, and Singapore MOE AcRF Tier 1 grant MOE2012-T1-001-094. Work partially done while at Georgia Institute of Technology, USA, and University of Vienna, Austria.Key words and phrases: algorithmic pricing, approximation algorithms, polynomial-time approximation scheme, bounded-treewidth graphs, bounded-genus graphs, sherali-adams hierarchy We study the Sherali-Adams hierarchy applied to a natural Integer Program formulation that (1 + ε)-approximates the optimal solution of GVP. Sherali-Adams hierarchy has gained much interest recently as a possible approach to develop new approximation algorithms. We show that, when the input graph has bounded treewidth or bounded genus, applying a constant number of rounds of Sherali-Adams hierarchy makes the integrality gap of this natural LP arbitrarily small, thus giving a (1 + ε)-approximate solution to the original GVP instance.On k-partite graphs, we present a constant-factor approximation algorithm. We further improve the approximation factors for paths, cycles and graphs with degree at most three.