[Paper Review] Graph Pricing Problem on Bounded Treewidth, Bounded Genus and k-partite graphs

Parinya Chalermsook, Shiva Kintali+2 more|arXiv (Cornell University)|Mar 8, 2012

Advanced Graph Theory Research14 references3 citations

TL;DR

This paper presents a fully polynomial-time approximation scheme (FPTAS) for the Graph Pricing Problem (GVP) on graphs with bounded treewidth and extends it to bounded genus and k-partite graphs. It leverages dynamic programming on tree decompositions and applies the Sherali-Adams hierarchy to reduce the integrality gap, achieving (1+ε)-approximation for bounded treewidth and genus graphs, while also providing a 2-approximation for graphs of maximum degree four.

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 {\em graph vertex pricing} ({\sf 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 {\sf FPTAS} for {\sf GVP} on graphs with bounded treewidth. This result is also extended to an {\sf FPTAS} for the more general {\em single-minded pricing} problem. On bounded genus graphs we present a {\sf PTAS} and show that {\sf GVP} is {\sf NP}-hard even on planar graphs. We study the Sherali-Adams hierarchy applied to a natural Integer Program formulation that $(1+ε)$-approximates the optimal solution of {\sf 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 {\sf LP} arbitrarily small, thus giving a $(1+ε)$-approximate solution to the original {\sf 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.

Motivation & Objective

• To develop efficient approximation algorithms for the Graph Pricing Problem (GVP) on special graph classes such as bounded treewidth, bounded genus, and k-partite graphs.
• To understand how the combinatorial structure of the input graph influences the approximability of GVP.
• To analyze the effectiveness of the Sherali-Adams hierarchy in reducing the integrality gap of a natural LP formulation for GVP.
• To establish hardness results, showing that GVP is NP-hard even on planar graphs (a subclass of bounded genus).
• To improve approximation factors for low-degree graphs, including paths, cycles, and graphs with maximum degree ≤4.

Proposed method

• Design a dynamic programming algorithm on tree decompositions to achieve an FPTAS for GVP on bounded treewidth graphs.
• Extend the FPTAS to bounded genus graphs using Baker’s and Eppstein’s techniques for planar and bounded-genus graph decomposition.
• Apply the Sherali-Adams hierarchy to a natural integer programming formulation of GVP, showing that a constant number of rounds reduces the integrality gap to (1+ε) for bounded treewidth and genus graphs.
• Use a decomposition technique to partition edges into two subgraphs of maximum degree two (paths and cycles), enabling polynomial-time optimal solutions on each, and combine them for a 2-approximation on degree-4 graphs.
• Formulate a linear program (LP_opt) to compute optimal prices when all edges contribute positive revenue, and use dynamic programming to solve paths and cycles.
• Derive a recursive algorithm for cycles by removing each edge and solving the resulting path, then taking the maximum revenue.

Experimental results

Research questions

• RQ1 Can an FPTAS be designed for the Graph Pricing Problem on graphs of bounded treewidth?
• RQ2 Does the Sherali-Adams hierarchy effectively reduce the integrality gap of the natural LP relaxation for GVP on bounded treewidth and bounded genus graphs?
• RQ3 What is the approximability of GVP on k-partite graphs, and can better approximation factors be achieved for special subcases like paths and cycles?
• RQ4 Is the Graph Pricing Problem NP-hard on planar graphs, a subclass of bounded genus?
• RQ5 Can a 2-approximation algorithm be achieved for graphs with maximum degree four using edge decomposition and subgraph optimization?

Key findings

• An FPTAS exists for the Graph Pricing Problem on graphs of bounded treewidth, achieved via dynamic programming on tree decompositions.
• The Sherali-Adams hierarchy reduces the integrality gap of the natural LP formulation to (1+ε) for bounded treewidth and bounded genus graphs after a constant number of rounds.
• A PTAS is developed for GVP on bounded genus graphs, and the problem is proven NP-hard even on planar graphs.
• A 2-approximation algorithm is presented for graphs with maximum degree four, using edge decomposition into two subgraphs of degree at most two.
• Polynomial-time algorithms are designed for paths and cycles, with optimal solutions computed via linear programming or dynamic programming.
• For graphs of degree at most three, a constant-factor approximation with ratio αk/(k−1) < 4 is achieved, improving upon general graph bounds.