A Quadratic Programming Relaxation Approach to Compute-and-Forward   Network Coding Design
  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. 
Baojian Zhou, Jinming Wen and Wai Ho Mow
cs.IT math.IT