Graph Pricing Problem on Bounded Treewidth, Bounded Genus and k-partite graphs

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 $(1+\epsilon)$-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 $(1+\epsilon)$-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.

💡 Research Summary

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The paper studies the Graph Vertex Pricing (GVP) problem, where a seller assigns prices to an unlimited supply of n products (vertices) and m consumers are represented by weighted edges, each wanting to buy the two incident products if the sum of their prices does not exceed the consumer’s budget. The goal is to maximize total revenue. While the problem is known to admit only an O(log n)‑approximation on general graphs, the authors focus on three important graph families—bounded treewidth, bounded genus (including planar), and k‑partite graphs—and obtain substantially stronger results.

Bounded‑treewidth graphs.
Using a tree‑decomposition of width w, the authors design a dynamic programming (DP) algorithm that works on the decomposition tree. Prices are discretized to multiples of Δ = ε·P_max / n, limiting each vertex to O(1/ε) candidate prices. For each bag of the decomposition (size ≤ w + 1) the DP stores the optimal revenue for every combination of candidate prices, and transitions enforce consistency across adjacent bags. The DP runs in O(n·(1/ε)^{w+1}) time, which is polynomial when w is a constant, yielding a Fully Polynomial‑Time Approximation Scheme (FPTAS). The same technique extends to the more general single‑minded pricing problem, where each consumer may desire an arbitrary subset of items.

Bounded‑genus graphs.
For graphs embeddable on a surface of genus g, the authors adapt Baker’s layering technique. The graph is partitioned into layers via a BFS ordering; removing every k‑th layer (k = ⌈1/ε⌉) leaves a subgraph whose treewidth is O(k·g). Applying the FPTAS on each such subgraph gives a (1 + ε)‑approximation for the original instance, because the removed layers contribute at most an ε‑fraction of the optimal revenue. This yields a Polynomial‑Time Approximation Scheme (PTAS) for any fixed genus. Moreover, by a reduction from Max‑Cut, the paper proves that GVP remains NP‑hard even on planar graphs (genus 0).

Sherali‑Adams hierarchy.
The natural integer program for GVP uses variables p_v (price of vertex v) and y_e (indicating whether edge e is satisfied). Its linear relaxation has a large integrality gap. The authors apply the Sherali‑Adams lift‑and‑project procedure. They show that after O(w·log 1/ε) rounds for treewidth w, or O(g·log 1/ε) rounds for genus g, the resulting LP has an integrality gap of at most 1 + ε. Consequently, solving this strengthened LP yields a (1 + ε)‑approximate solution without the need for the DP, highlighting the power of a constant‑level hierarchy on structurally restricted graphs.

k‑partite graphs.
When the input graph is k‑partite, the authors propose a simple yet effective algorithm: they assign prices independently within each part and then adjust prices on edges crossing parts to ensure that at least one endpoint’s price is zero, thus guaranteeing that the edge’s contribution is counted. This yields an approximation factor of 4·(k − 1)/k. For the special cases of paths and cycles (2‑partite), a dynamic program computes an optimal or a 2‑approximation solution. For graphs of maximum degree three, a matching‑based rounding achieves a 3‑approximation.

Overall contribution.
The work demonstrates that structural graph properties dramatically improve the approximability of GVP. Bounded treewidth admits an FPTAS via DP and price discretization; bounded genus admits a PTAS via Baker‑style layering; a modest number of Sherali‑Adams rounds close the LP gap on these families; and k‑partite and low‑degree graphs admit constant‑factor algorithms with refined analyses for paths, cycles, and degree‑three graphs. The results not only settle several open questions about the hardness of GVP on planar graphs but also provide a template for leveraging graph decompositions and LP hierarchies in other combinatorial pricing problems. Future directions include extending these techniques to minor‑free graphs, online or stochastic pricing settings, and empirical evaluation on real‑world marketplaces.