# Polynomial-Time Methods to Solve Unimodular Quadratic Programs With Performance Guarantees

## TL;DR

This paper introduces three polynomial-time heuristic algorithms for approximately solving NP-hard unimodular quadratic programs, with performance guarantees and applications in sensing and wireless communication.

## Contribution

The paper proposes three novel heuristic methods for UQPs, including dominant-eigenvector-matching, greedy, and row-swap strategies, with theoretical performance guarantees.

## Key findings

- Dominant-eigenvector-matching method has a proven performance guarantee.
- Greedy strategy achieves (1-1/e) approximation ratio under string submodularity.
- Numerical results show these heuristics outperform standard semidefinite relaxation.

## Abstract

We develop polynomial-time heuristic methods to solve unimodular quadratic programs (UQPs) approximately, which are known to be NP-hard. In the UQP framework, we maximize a quadratic function of a vector of complex variables with unit modulus. Several problems in active sensing and wireless communication applications boil down to UQP. With this motivation, we present three new heuristic methods with polynomial-time complexity to solve the UQP approximately. The first method is called dominant-eigenvector-matching; here the solution is picked that matches the complex arguments of the dominant eigenvector of the Hermitian matrix in the UQP formulation. We also provide a performance guarantee for this method. The second method, a greedy strategy, is shown to provide a performance guarantee of (1-1/e) with respect to the optimal objective value given that the objective function possesses a property called string submodularity. The third heuristic method is called row-swap greedy strategy, which is an extension to the greedy strategy and utilizes certain properties of the UQP to provide a better performance than the greedy strategy at the expense of an increase in computational complexity. We present numerical results to demonstrate the performance of these heuristic methods, and also compare the performance of these methods against a standard heuristic method called semidefinite relaxation.