On the Approximability of Independent Set Problem on Power Law Graphs

TL;DR

This paper establishes new nonconstant lower bounds for approximating the Independent Set Problem on Power Law Graphs, revealing limitations in algorithmic performance depending on the power law exponent.

Contribution

It provides the first nonconstant lower bounds for approximation on Power Law Graphs, using a novel embedding technique applicable when the exponent is less than or equal to one.

Findings

Lower bounds of $n^{ ext{ extepsilon}}$ for $eta<1$

Lower bounds of $ ext{ extlog}(n)^{ ext{ extepsilon}}$ for $eta=1$

Embedding technique of independent interest

Abstract

We give the first nonconstant lower bounds for the approximability of the Independent Set Problem on the Power Law Graphs. These bounds are of the form $n^{\epsilon}$ in the case when the power law exponent satisfies $\beta <1$. In the case when $\beta =1$, the lower bound is of the form $\log (n)^{\epsilon}$. The embedding technique used in the proof could also be of independent interest.