A note on anti-coordination and social interactions

TL;DR

This paper investigates the computational complexity of the maximum independent cut problem, a variant of MAX-CUT with an independent set constraint, proving it is hard to approximate within certain bounds.

Contribution

It confirms a conjecture that the problem is NP-hard to approximate within any ratio better than $n^{1- ext{epsilon}}$, even for graphs with maximum degree four.

Findings

No polynomial-time approximation better than $n^{1- ext{epsilon}}$ unless P=NP.

The problem remains MAXSNP-hard for graphs with degree at most four.

Supports the conjecture with complexity and hardness results.

Abstract

This note confirms a conjecture of [Bramoull\'{e}, Anti-coordination and social interactions, Games and Economic Behavior, 58, 2007: 30-49]. The problem, which we name the maximum independent cut problem, is a restricted version of the MAX-CUT problem, requiring one side of the cut to be an independent set. We show that the maximum independent cut problem does not admit any polynomial time algorithm with approximation ratio better than $n^{1-\epsilon}$, where $n$ is the number of nodes, and $\epsilon$ arbitrarily small, unless P=NP. For the rather special case where each node has a degree of at most four, the problem is still MAXSNP-hard.