Analysis of Precoding-Based Intersession Network Coding and The Corresponding 3-Unicast Interference Alignment Scheme Jaemin Han, Chih-Chun Wang Center of Wireless Systems and Applications (CWSA) School of Electrical and Computer Engineering, Purdue University Email: {han83,chihw}@purdue.edu Ness B. Shroff Department of ECE and CSE The Ohio State University Email: shroff@ece.osu.edu Abstract—Recently, a new precoding-based intersession net- work coding (NC) scheme has been proposed, which applies the interference alignment technique, originally devised for wireless interference channels, to the 3-unicast problem of directed acyclic networks. Motivated by the graph-theoretic characterizations of classic linear NC results, this paper investigates several key relationships between the point-to-point network channel gains and the underlying graph structure. Such relationships are critical when characterizing graph-theoretically the feasibility of precoding-based solutions. One example of the applications of our results is to answer (at least partially) the conjectures of the 3- unicast interference alignment technique and the corresponding graph-theoretic characterization conditions. Index Terms—Asymptotic interference alignment, interference channels, intersession network coding, 3-unicast networks. I. I NTRODUCTION Characterizing the capacity or the feasibility of satisfying the network traffic demands of multiple coexisting source- destination pairs (sessions) has been a long-standing challenge. Recently, by allowing the coding operation to be performed at the intermediate network nodes, a new concept of network coding has emerged, which is able to achieve the information- theoretic capacity for the single multicast [2] even when considering only linear network codes [3]. Several papers have since studied the network code construction problem for the above single multicast setting [4]–[7]. On the other hand, when there are multiple coexisting sessions in the network, the corresponding network code de- sign/analysis problem, also known as the intersession network coding (INC) problem, becomes notoriously challenging due to the potential interference within the network. For example, linear network coding no longer achieves the capacity [10]. Deciding the existence of a (linear) network code that satisfies general traffic demands becomes an NP-hard problem [5], [9]. Thus, recent INC studies have focused on the optimal characterizations over some restrictive networks or limited rate constraints, including the capacity regions for directed cycles [14], degree-2 three-layer directed acyclic networks (DAG) [15], and for networks with integer link capacity and two coexisting rate-1 multicast sessions [8]. Recently, the authors in [12], [13] applied the interference alignment (IA) technique, originally developed for wireless interference channels [11], to the scenario of 3 coexisting unicast sessions in the name of 3-unicast Asymptotic Net- work Alignment (ANA). Their application of the interference- aligning idea leads to a new perspective to the INC problems, which enables us to focus on designing the precoding and decoding mappings at the sources and destinations while allowing randomly generated local encoding kernels [7] within the network. Compared to the classic algebraic framework that fully controls the encoder, decoder, and local encoding kernels [5], this precoding-based framework tradeoffs the ultimate achievable throughput with a distributed, implementation- friendly structure that allows pure random linear NC in the interior of the network. Their initial study on 3-unicast net- works shows that, by performing precoding across multiple time slots and applying the IA technique, the precoding-based NC can perform strictly better than the pure routing solution in some networks, and even better than some widely-used linear NC solutions in some networks. Such results thus demonstrate a new balance between practicality and throughput enhance- ment. Further development of the precoding-based framework could thus have significant impact on practical network code design. In this work, we first study several basic properties of the precoding-based framework, and then apply our results to the 3-unicast ANA scheme proposed in [12], [13], which applies the interference alignment (IA) technique to the 3-unicast net- work. The existing results [12], [13] show that when the net- work code satisfies certain algebraic conditions, the 3-unicast ANA scheme achieves asymptotically half of the interference- free throughput for each transmission pair. Note that for the wireless interference channels where the IA technique was originally developed, those algebraic feasibility conditions can be satisfied with close-to-one probability provided the channel gains are independent and continuously distributed random variables [11]. For comparison, the “network channel gains” are generally correlated and the correlation depends heavily on the underlying network topology [12], [13]. As a result, we need new efficient ways to decide whether the network of interest admits a 3-unicast ANA scheme that achieves half of the interference-free throughput. The results in this work answer this question by developing new graph-theoretic conditions which are equivalent (at least partially) to the feasibility of the 3-unicast ANA scheme. The proposed graph-