International Symposium on Information Theory and its Applications, ISITA2004 Parma, Italy, October 10–13, 2004 Network Coding with a Cost Criterion Desmond S. Lun † , Muriel M´ edard † , Tracey Ho † , and Ralf Koetter ‡ † Laboratory for Information and Decision Systems Massachusetts Institute of Technology Cambridge, MA 02139, USA E-mail: {dslun, medard, trace}@mit.edu ‡ Coordinated Science Laboratory University of Illinois at Urbana-Champaign Urbana, IL 61801, USA E-mail: koetter@uiuc.edu Abstract We consider applying network coding in settings where there is a cost associated with network use. We show that, while minimum-cost multicast problems without network coding are very difficult except in the special cases of unicast and broadcast, finding minimum-cost subgraphs for single multicast connections with net- workcodingcanbeposedasalinearoptimizationprob- lem. In particular, we apply our approach to the prob- lem of minimum-energy multicast in wireless networks with omnidirectional antennas and show that it can be handled by a linear optimization problem when net- work coding is used. For the case of multiple multicast connections, we give a partial solution: We specify a linear optimization problem that yields a solution of no greater cost than any solution without network coding and that we suspect can potentially be substantially better. 1. Introduction The selection of routes is an issue of utmost impor- tanceindatanetworksthathassofarreceivedscantat- tention in the literature on network coding. Indeed, the standard framework in which network coding is cast, that of network information flow problems [2], assumes that we have a network with limited-capacity links and considers whether or not a given set of connections can be simultaneously established, but gives no consider- ation to the resources that are consumed as a result of communicating on the links. In addition, such a framework implicitly assumes a certain homogeneity in network traffic — the goal is to ensure that connec- tions are established as long as the network has the capacity to accommodate them, regardless of the type or purpose of the connections — which is frequently This work was supported by the National Science Founda- tion under grant nos. ANI-0335256 and CCR-0325673, and by Hewlett-Packard under grant no. 008542-008. not the case. The most notable example is today’s in- ternet, which not only carries different types of traffic, but is also used by a vastly heterogeneous group of end users with differing valuations of network service and performance. It has been variously proposed that such heterogeneous networks be priced [20], with some mod- els allowing for selfish routing decisions based on the price of the links [13, 1]. In the present paper, we consider applying network coding in settings where there is a cost associated with network use, our natural objective being to select sub- graphs for coding that minimize the cost incurred. We commence by considering single multicast con- nections (which include single unicast and broadcast connectionsasspecialcases)inthefollowingsection. In Section 3, we study a particular instance of minimum- cost single multicast connections that has attracted much recent interest, that of minimum-energy multi- cast in wireless networks. In Section 4, we treat the case of multiple multicast connections. 2. Single multicast connections Whenever the members of a multicast group have a selfish cost objective, or when the network sets link weights to meet its objective or enforce certain policies and each multicast group is subject to a minimum- weight objective, we wish to set up single multicast connections at minimum cost. Network coding for sin- gle multicast connections is relatively simple as we have a simple characterization of feasibility in networks with limited-capacity links [2, Theorem 1] and, moreover, it is known that it suffices to consider linear operations over a sufficiently large finite field on a sufficiently long vector created from the source process [15, Theorem 3.3], [14, Theorem 4]. We model the network with a directed graph G = (N,A). For each link (i,j ) ∈ A, we associate non- negative numbers a ij and c ij , which are the cost per unit flow and the capacity of the link, respectively.