On the Approximation Resistance of Balanced Linear Threshold Functions

TL;DR

This paper demonstrates that certain balanced linear threshold functions are hard to approximate, challenging previous conjectures, while also showing that the almost monarchy predicate becomes approximable as the number of variables increases.

Contribution

It proves the approximation resistance of some balanced linear threshold functions and establishes the approximability of the almost monarchy predicate for large k.

Findings

Existence of a balanced LTF that is unique games hard to approximate.

Refutation of a previous conjecture about approximation resistance.

Almost monarchy predicate is approximable for sufficiently large k.

Abstract

In this paper, we show that there exists a balanced linear threshold function (LTF) which is unique games hard to approximate, refuting a conjecture of Austrin, Benabbas, and Magen. We also show that the almost monarchy predicate on k variables is approximable for sufficiently large k.