Published by the Department of Computer Science, The University of Chicago.

Parinya Chalermsook

Max Planck-Institut für Informatik

Saarbrücken, Germany

`painya AT cs DOT uchicago DOT edu `

Shiva Kintali

Department of Computer Science

Princeton University

Princeton, NJ-08540

USA

`kintali AT cs DOT princeton DOT edu`

Richard J. Lipton

College of Computing

Georgia Institute of Technology

Atlanta GA-30332

USA

`rjl AT gatech DOT edu`

and

Danupon Nanongkai

School of Physical and Mathematical Sciences

Nanyang Technological University

Singapore

`danupon AT gmail DOT com`

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.

We study the Sherali-Adams hierarchy applied to a natural Integer Program formulation that $(1+\epsilon)$-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 $(1+\epsilon)$-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.

Submitted July 3, 2012, revised September 9, 2013 and November 7, 2013;
published November 9, 2013.

© 2013 Parinya Chalermsook, Shiva Kintali, Richard J. Lipton, and Danupon Nanongkai

Licensed under a Creative Commons Attribution License

Licensed under a Creative Commons Attribution License

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