Approximation Complexity of Max-Cut on Power Law Graphs

TL;DR

This paper investigates the approximability of the MAX-CUT problem on power law graphs, establishing polynomial-time approximation schemes for certain exponents and hardness results for others, along with improved algorithms.

Contribution

It provides the first PTAS for MAX-CUT on power law graphs with exponent in (0,2) and proves NP-hardness for approximation within any ratio greater than 1 for exponents above 2.

Findings

PTAS exists for eta in (0,2)

NP-hardness for eta > 2

Improved approximation algorithms for eta > 2

Abstract

In this paper we study the MAX-CUT problem on power law graphs (PLGs) with power law exponent $\beta$. We prove some new approximability results on that problem. In particular we show that there exist polynomial time approximation schemes (PTAS) for MAX-CUT on PLGs for the power law exponent $\beta$ in the interval $(0,2)$. For $\beta>2$ we show that for some $\epsilon>0$, MAX-CUT is NP-hard to approximate within approximation ratio $1+\epsilon$, ruling out the existence of a PTAS in this case. Moreover we give an approximation algorithm with improved constant approximation ratio for the case of $\beta>2$.