A sufficient condition for interference alignment

We consider the classical problem of spatial interference alignment (IA) in MIMO channels with constant channel coefficients through design of linear transmit precoders and receiver filters. Some easily (polynomial time) computable necessary conditions for IA have been derived in the literature [1], [2], [3]. Computable sufficient and necessary conditions that completely characterizes the feasibility of an IA problem have also been obtained [4], [2]. However, it has been shown that checking the feasibility of interference alignment when the number of antennas are more than two is NP-complete[3]. This result is inline with full characterization of the feasibility of IA as the sufficiency conditions require multiplication of Schubert cycles that becomes exhaustive as the dimensions grows. Naturally, the following questions may arise: “Is it possible to have a sufficiency condition for a general case of IA based on only the dimensions of the system (number of antennas at each node and degrees of freedom (DoF) per node [4]) that is simple (polynomial time) to compute?” and “How effective such sufficiency conditions would be?”. In this paper, we provide an affirmative answer to the first question and show the proposed sufficient condition is asymptotically optimal. The sufficiency conditions are expressed in terms of simple inequalities based on system dimensions. Unlike necessary conditions that are based on simple argument such as dimension counting [1], we have not yet been able to provide an elementary proof for the derived sufficiency conditions. The provided proof requires familiarity with Schubert calculus over complex Grassmannians.