Random Linear Intersession Network Coding With Selective Cancelling Chih-Chun Wang Center of Wireless Systems and Applications (CWSA) School of ECE, Purdue University Ness B. Shroff Departments of ECE and CSE The Ohio State University Abstract—The network coding capacity of a single multicast traffic is characterized by the min-cut/max-flow (mcMF) theorem, which can be achieved by random linear network coding (RLNC). Nonetheless, the graph-theoretic characterization for multiple unicast/multicast traffic remains an open problem. This paper proposes and studies a new class of intersession-network-coding schemes: RLNC with selective cancelling (SC), which inherits the complexity advantage of RLNC once the set of selective cancelling edges is decided. A graph-theoretic characterization is provided for the achievable rates of RLNC with SC for the general multiple multicast setting. The findings contain most existing achievability results as special cases, including the mcMF theorem of the single multicast traffic and the existing characterization of pairwise intersession network coding. One prominent feature of the proposed approach is its focus on the achievability analysis with arbitrary network topology, arbitrary inter-session packet-mixing capability, and arbitrary traffic demands, which distinguishes the results from the special case analysis, capacity outer bound constructions, and the pattern-based (butterfly- based) superposition arguments. I. I NTRODUCTION Network coding allows packet mixing at intermediate nodes and provides strict throughput improvements for modern com- munication networks. The capacity region of network coding is well understood when only a single multicast session exists in the network [1]. Network coding across multiple sessions, termed intersession network coding, also demonstrates signif- icant throughput improvement in the butterfly network [2] and empirically enhances the network performance by 50–200% [3]. Nonetheless, the problem of a unifying characterization of intersession network coding remains largely open. The lack of progress on intersession network coding is mainly attributed to its intrinsic hardness. For example, check- ing the existence of intersession network coding solutions is an NP-complete problem [4], its information-theoretic charac- terization is heavily intertwined with the notoriously elusive fundamental regions of the entropy function [5], and linear coding is insufficient to achieve the intersession coding ca- pacity [6]. Even when restricting our focus to only linear codes on directed acyclic networks, it is shown [7], [8] that the feasibility of linear intersession network coding is alphabet GF(q) dependent, and the complexity of determining the feasibility of linear intersession network coding is similar to that of the long standing problems of finding solutions of multi-variate polynomials. It is worth mentioning that with a fixed number of N coexisting sessions and a fixed alphabet size GF(q), finding a network coding solution is a polynomial- time task with respect to the network size [9]. However, the complexity grows exponentially with respect to N and b = log 2 (q), the number of bits representing each alphabet. To better understand the capacity of intersession net- work coding, many ongoing works focus on more tractable outer/inner bounds analysis for networks of general topology. Capacity outer bounds are often devised based on generalized cut conditions (see [10] and the references therein). For the achievable rate regions (capacity inner bounds), existing works can be categorized into three major approaches. Since the butterfly network is well-known to admit intersession coding benefits, many works focus on decomposing the orig- inal network into many butterfly substructures [11] and use linear programming for the corresponding network resource allocation [12]. The second type of approaches classifies the coded traffic by the participating sessions. The traffic flow that is a mixture of N ′ unicast sessions can then be treated as generated from a single multicast session with N ′ sym- bols, to which one can apply the min-cut/max-flow (mcMF) theorem [13], [14]. Recently, a new characterization has been discovered for pairwise intersession network coding, which mixes only two symbols of two coexisting unicast/multicast sessions [2]. When combined with the superposition principle, the pairwise coding results are used to derive new achievable rates for N coexisting sessions [15]. This paper proposes and studies a new class of intersession- network-coding schemes: random linear network coding (RLNC) with selective cancelling (SC) for networks of general topology. A graph-theoretic characterization is provided for the feasibility of RLNC w. SC in the setting of N multicast ses- sions. The results include as special cases all existing achiev- ability approaches and characterize a strictly larger achievable rate region. Although focusing only on achievable rates, RLNC w. SC is capable of achieving the capacity in many instances including the setting of a single multicast session. This work can thus be viewed as a generalization of the achievability part of the mcMF theorem from the single multicast session to multiple multicasts. The random construction of our scheme inherits the complexity advantages of RLNC [16] once the set of selective cancelling edges is decided. With the complexity advantage and the larger achievable throughput, RLNC w. SC could have impact on designing high-performance intersession network coding protocols. 2009 IEEE Information Theory Workshop 978-1-4244-4983-5/09/$25.00 © 2009 IEEE 559 Authorized licensed use limited to: The Ohio State University. Downloaded on June 02,2010 at 21:40:43 UTC from IEEE Xplore. Restrictions apply.