Hardness of Graph Pricing through Generalized Max-Dicut

TL;DR

This paper establishes the NP-hardness of approximating the Graph Pricing problem within a factor better than 1/4 under UGC, and introduces a new CSP called Generalized Max-Dicut to analyze its complexity.

Contribution

It introduces Generalized Max-Dicut, links it to Graph Pricing via reduction, and proves tight hardness and integrality gap results for these problems.

Findings

NP-hardness of approximating Graph Pricing within 1/4 under UGC

Existence of integrality gap at most 1/2 + ε for certain LP relaxations

Reduction from Generalized Max-Dicut to Graph Pricing

Abstract

The Graph Pricing problem is among the fundamental problems whose approximability is not well-understood. While there is a simple combinatorial 1/4-approximation algorithm, the best hardness result remains at 1/2 assuming the Unique Games Conjecture (UGC). We show that it is NP-hard to approximate within a factor better than 1/4 under the UGC, so that the simple combinatorial algorithm might be the best possible. We also prove that for any $\epsilon > 0$, there exists $\delta > 0$ such that the integrality gap of $n^{\delta}$-rounds of the Sherali-Adams hierarchy of linear programming for Graph Pricing is at most 1/2 + $\epsilon$. This work is based on the effort to view the Graph Pricing problem as a Constraint Satisfaction Problem (CSP) simpler than the standard and complicated formulation. We propose the problem called Generalized Max-Dicut($T$), which has a domain size $T + 1$ for every $T \geq 1$. Generalized Max-Dicut(1) is well-known Max-Dicut. There is an approximation-preserving reduction from Generalized Max-Dicut on directed acyclic graphs (DAGs) to Graph Pricing, and both our results are achieved through this reduction. Besides its connection to Graph Pricing, the hardness of Generalized Max-Dicut is interesting in its own right since in most arity two CSPs studied in the literature, SDP-based algorithms perform better than LP-based or combinatorial algorithms --- for this arity two CSP, a simple combinatorial algorithm does the best.