A polynomial time algorithm for a variant of shortest vector problem with applications in compute-and-forward design

Abstract

In wireless communication networks, one of the most significant problems in Compute-and-Forward (CPF) strategy based on Integer-Forcing (IF) technique is to search the optimal integer coefficient vector for a relay to decode the received messages. While this is a variant of shortest vector problem (SVP), which is prohibitively complex in its general form. In this paper, we mainly focus on the optimal integer solution with non-zero entries in multi-source one-relay model. For comparison we extend the number (or dimension) of communication nodes from 2 to any larger integer n based on an existing 2-dimensional search algorithm with non-zero entries. Then we add this non-zero-entries constraint to an existing search algorithm and make some improvements to search the optimal integer vector by solving an optimization problem over only one variable instead of searching all vector components, which has a much lower time complexity. We prove that our new search algorithm is more efficient with polynomial complexity compared with the extended n-dimensional algorithm, meanwhile the same optimal integer solution is guaranteed.