Improved Approximation Algorithms for Orthogonally Constrained Problems Using Semidefinite Optimization

TL;DR

This paper develops a polynomial-time approximation algorithm for orthogonally constrained quadratic problems, extending semidefinite relaxation techniques and providing tight approximation guarantees similar to those in Max-Cut.

Contribution

It introduces a new semidefinite relaxation and randomized rounding method for orthogonally constrained problems, achieving a proven 1/3-approximation ratio.

Findings

Achieves a 1/3-approximation ratio for the problem.

Constructs instances where the approximation ratio approaches 1/3.

Extends semidefinite optimization techniques to orthogonally constrained problems.

Abstract

Building on the blueprint from Goemans and Williamson (1995) for the Max-Cut problem, we construct a polynomial-time approximation algorithm for orthogonally constrained quadratic optimization problems. First, we derive a semidefinite relaxation and propose a randomized rounding algorithm to generate feasible solutions from the relaxation. Second, we derive constant-factor approximation guarantees for our algorithm. When optimizing for $m$ orthonormal vectors in dimension $n$, we leverage strong duality and semidefinite complementary slackness to show that our algorithm achieves a $1/3$-approximation ratio. For any $m$ of the form $2^q$ for some integer $q$, we also construct an instance where the performance of our algorithm is exactly $(m+2)/(3m)$, which can be made arbitrarily close to $1/3$ by taking $m \rightarrow + \infty$, hence showing that our analysis is tight.