A Quadratic Programming Relaxation Approach to Compute-and-Forward Network Coding Design

TL;DR

This paper introduces a quadratic programming relaxation method for optimizing integer coefficients in compute-and-forward network coding, achieving high rates with low computational complexity.

Contribution

It proposes a novel relaxation approach transforming a complex integer optimization into quadratic programs with closed-form solutions, improving efficiency over existing methods.

Findings

Achieves comparable rates to existing methods

Offers a low-complexity solution for coefficient optimization

Demonstrates effectiveness through numerical experiments

Abstract

Using physical layer network coding, compute-and-forward is a promising relaying scheme that effectively exploits the interference between users and thus achieves high rates. In this paper, we consider the problem of finding the optimal integer-valued coefficient vector for a relay in the compute-and-forward scheme to maximize the computation rate at that relay. Although this problem turns out to be a shortest vector problem, which is suspected to be NP-hard, we show that it can be relaxed to a series of equality-constrained quadratic programmings. The solutions of the relaxed problems serve as real-valued approximations of the optimal coefficient vector, and are quantized to a set of integer-valued vectors, from which a coefficient vector is selected. The key to the efficiency of our method is that the closed-form expressions of the real-valued approximations can be derived with the Lagrange multiplier method. Numerical results demonstrate that compared with the existing methods, our method offers comparable rates at an impressively low complexity.