Maximum Quadratic Assignment Problem: Reduction from Maximum Label Cover and LP-based Approximation Algorithm

TL;DR

This paper establishes hardness results for approximating the maximum quadratic assignment problem and related problems, and introduces an LP-based approximation algorithm with an $O( sqrt{n})$ ratio.

Contribution

It proves strong inapproximability bounds for the maximum quadratic assignment problem and related problems, and proposes an LP relaxation rounding algorithm with a specific approximation ratio.

Findings

Hardness of approximation within $2^{ ext{log}^{1- ext{epsilon}} n}$ factor unless NP $ extsubseteq$ BPQP.

Approximate Graph Isomorphism is hard assuming the Unique Games Conjecture.

An $O( sqrt{n})$-approximation algorithm based on LP relaxation rounding.

Abstract

We show that for every positive $\epsilon > 0$, unless NP $\subset$ BPQP, it is impossible to approximate the maximum quadratic assignment problem within a factor better than $2^{\log^{1-\epsilon} n}$ by a reduction from the maximum label cover problem. Our result also implies that Approximate Graph Isomorphism is not robust and is in fact, $1 - \epsilon$ vs $\epsilon$ hard assuming the Unique Games Conjecture. Then, we present an $O(\sqrt{n})$-approximation algorithm for the problem based on rounding of the linear programming relaxation often used in the state of the art exact algorithms.