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Sparsest cut on bounded treewidth graphs: Algorithms and hardness results
 In 45th Annual ACM Symposium on Symposium on Theory of Computing
, 2013
"... We give a 2approximation algorithm for NonUniform Sparsest Cut that runs in time nO(k), where k is the treewidth of the graph. This improves on the previous 22 kapproximation in time poly(n)2O(k) due to Chlamtác ̌ et al. [CKR10]. To complement this algorithm, we show the following hardness resul ..."
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We give a 2approximation algorithm for NonUniform Sparsest Cut that runs in time nO(k), where k is the treewidth of the graph. This improves on the previous 22 kapproximation in time poly(n)2O(k) due to Chlamtác ̌ et al. [CKR10]. To complement this algorithm, we show the following hardness results: If the NonUniform Sparsest Cut problem has a ρapproximation for seriesparallel graphs (where ρ ≥ 1), then the MaxCut problem has an algorithm with approximation factor arbitrarily close to 1/ρ. Hence, even for such restricted graphs (which have treewidth 2), the Sparsest Cut problem is NPhard to approximate better than 17/16 − ε for ε> 0; assuming the Unique Games Conjecture the hardness becomes 1/αGW − ε. For graphs with large (but constant) treewidth, we show a hardness result of 2 − ε assuming the Unique Games Conjecture. Our algorithm rounds a linear program based on (a subset of) the SheraliAdams lift of the standard Sparsest Cut LP. We show that even for treewidth2 graphs, the LP has an integrality gap close to 2 even after polynomially many rounds of SheraliAdams. Hence our approach cannot be improved even on such restricted graphs without using a stronger relaxation. 1
Graph Pricing Problem on Bounded Treewidth, Bounded Genus and kPartite Graphs
, 2013
"... 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 wa ..."
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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 kpartite 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 singleminded pricing problem. On bounded genus graphs we present a PTAS and show that GVP is NPhard even on planar graphs.