# The Encoding Complexity of Network Coding

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5 Although the problem of finding the exact or approximate value of 132 is -hard, we upper bounds on 132 that hold for any instance of the multicast network coding problem. The main contribution of our paper is an upper bound on 132 which is independent of the size of the network and depends on and % only. Specifically, we show that 12 for any acyclic coding network that delivers packets to terminals. Our bound is constructive, i.e., for any feasible instance we present an efficient procedure that constructs a network code with at most encoding nodes. In what follows, an algorithm is said to be efficient if its running time is polynomial in the size of the underlying graph. Theorem 4 (Upper bound, acyclic networks) Let be an acyclic graph and let be a feasible instance of the multicast network coding problem. Then, one can efficiently find a feasible network code for with at most encoding nodes, i.e., 12, where %. Theorem 5 (Lower bound, acyclic networks) Let and be arbitrary integers. Then, there exist instances of the network coding problem such that, %, the underlying graph is acyclic, and 12. Finally, we establish upper and lower bounds on the number of encoding nodes in the general setting of communication networks with cycles. We show that the value of 132 in a cyclic network depends on the size of the minimum feedback link set. Definition 6 (Minimum feedback link set [10]) Let be a directed graph. A subset is referred to as a feedback link set if the graph - formed from by removing all links in is acyclic. A feedback link set of minimum size is referred to as the minimum feedback link set. Given a network, we denote by the minimum feedback link set of its underlying graph. Theorem 7 (Upper bound, cyclic networks) Let be an instance of the multicast network coding problem. Then, one can efficiently find a feasible network code for with at most 1 encoding nodes, i.e., 12 1, where %. Theorem 8 (Lower bound, cyclic networks) Let and be arbitrary integers. Then (a) there exists instances of the multicast network coding problem such that, and 132 (b) there exist instances of the multicast network coding problem such that, %, and < (here is the set of nodes in ). 132 A couple of remarks are in place. First, note that Theorem 7 generalizes Theorem 4, as for acyclic networks the minimum feedback link set is of size 0. Second, note that Theorem 8 establishes two lower bounds. The first complements the upper bound of Theorem 7, while the second shows that in the case of cyclic networks the value of 132 is not necessarily independent of the size of the network and may depend linearly on the number of nodes in. 3 Simple coding networks Let be a feasible instance to the network coding problem. In order to establish a constructive upper bound on the minimal number of encoding nodes 132 of we consider a special family of feasible networks, referred to as simple networks. In what follows we define simple coding networks, and show that finding network codes with few encoding nodes for this family of restricted networks suffices to prove Theorems 4 and 7. We start by defining feasible instances which are minimal with respect to link removal. 5

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