Network Coding for Multiple Unicasts: An Approach based on Linear Optimization

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1 Network Coding for Mltiple Unicasts: An Approach based on Linear Optimization Danail Traskov, Niranjan Ratnakar, Desmond S. Ln, Ralf Koetter, and Mriel Médard Abstract In this paper we consider the application of network coding to a mltiple nicast setp. We present two sboptimal, yet practical code constrction techniqes. One consists of a linear program and the other of an integer program with fewer variables and constraints. We discss the performance of the proposed techniqes as well as their complexity. I. INTRODUCTION We consider the problem of setting p several simltaneos and independent point-to-point connections that share a common network and refer to this as a mltiple nicast setp. The notion of network coding [1] permits intermediate network nodes to encode observed data as opposed to traditional operation, where the network nodes are restricted to roting and replicating. An example of sch an encoding is illstrated in Fig. 1(a), where observes b 1 and b 2 and transmits b 1 b 2. We are interested in applying network coding to the mltiple nicast setp. Most reslts on network coding so far assme a mlticast setting, i.e. a single sorce transmits the same data simltaneosly to a set of receivers [1, 2]. This scenario is considerably simpler than the mltiple nicast problem. One simplifying aspect is the fact that the problem of finding a minimm-cost mlticast connection decomposes into two independent sbproblems [3]. First, one has to identify a sbgraph whose link capacities can spport the mlticast connection. Second, the code can be constrcted independent of the sbgraph by e.g. a randomized code constrction [2]. On the other hand, in the mltiple nicast scenario, the problems of selecting a sbgraph and the code constrction have to be solved jointly. The selection of a sbgraph is typically modeled as an optimization problem on flows in a network, whereas the code constrction is an algebraic and combinatorial problem. Since we have to solve them jointly, we restrict orselves to simple network codes, whose constrction can be combined with network flow problems of tractable complexity. We assme the nderlying network to be a directed graph G =(V, E), where V and E are the sets of vertices and directed edges respectively. We desire commnication of independent information from nodes s i to d i at a rate R i for i =1,...,n. A connection is defined as a triple (s i,d i,r i ). The n triples (s i,d i,r i ) are feasible with network coding if there exists a network code that allows for simltaneos transmission of information at a rate R i from s i to d i. The triples are said to D. Traskov, N. Ratnakar, and R. Koetter are with the Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, 1308 Main St., Urbana IL ( {traskov2, ratnakar, koetter}@ic.ed). This material is based pon work spported by the NSF nder Award No. NSF CCR D. S. Ln, M. Médard are with the Laboratory for Information and Decision Systems, Massachsetts Institte of Technology, Cambridge, MA ( {dsln, medard}@mit.ed). be feasible with roting if there exists a network code involving only roting and in this case the problem of determining flows that satisfy these connections is referred to as a mlticommodity flow problem [4]. The application of network coding in a mltiple nicast setp is discssed in [5]. In particlar, we know of two approaches for constrcting network codes for the non-mlticast scenario. In [6], the athors present algebraic code constrction techniqes for all connection triples that are feasible with linear network coding. However, the comptational complexity of solving the reslting system of polynomial eqation is exponential in the size of the nderlying network and therefore hardly tractable in practice. On the other hand, the athors in [7] present an opportnistic heristic code constrction for wireless networks. The approach presented in this paper fits in between these two schemes. We present code constrction techniqes for certain connection triples that are feasible with network coding, bt are not necessarily feasible with roting. However, we emphasize that the techniqes described here fail to provide network codes for all connection triples that are feasible with linear network coding. We restrict network coding to operations sing only binary XOR, derive flow formlations of the problem, and state it first as a linear program and then as an integer program. This constittes a sboptimal, yet practical and systematic scheme to constrct network codes. The linear and integer program model the same problem and, althogh not eqivalent, are qite similar. Their main difference lies in comptational isses. The paper is organized as follows: In Section II, we present a code constrction scheme based on a linear program. In Section III we present an alternative formlation with fewer variables and constraints, albeit in the form of an integer program. A comparison of both formlations in terms of their complexity and simlation reslts is given in Section IV. We smmarize or reslts in Section V. II. CODE CONSTRUCTION AS A LINEAR PROGRAM For an edge e =(, v) we define tail(e)=and head(e)=v. We se Γ I (v) to denote the set of edges whose head is v V and Γ O (v) to denote the set of edges with tail v. The n triples (s i,d i,r i ) are feasible with roting if the following conditions are satisfied [4]. For i =1, 2,...,nand e E, v V, n x e (i) z e C e, x e (i) 0 (1) x e (i) R i if v = s i x e (i) = R i if v = d i 0 otherwise (2) /06/$ IEEE 1758

2 s 1 s b 1 b b 1 b 2 b 1 d b 1 b 2 b 1 b 2 (a) d 1 b 2 r(1 2) = 1 s 1 s 2 d 2 q(1 2) = 1 q(2 1) = 1 p(1 2) = 1 p(2 1) = 1 p(1 2) = 1p(2 1) = 1 Fig. 1. (a) shows the btterfly network. The edges 1 throgh 7 are marked and the network coding soltion with the corresponding transmission on each edge is shown. b i represents data for the i-th connection for i =1, 2. In (b), the btterfly network with the poison, antidote reqest, and antidote variables that are non-zero on each edge. All variables correspond to the poison generated at node. In the above eqations, x e (i) is a flow from s i to d i of vale R i and z e is the aggregate load on e. However, when network coding is permitted, the notion of a flow is not clear. For example, consider the so-called btterfly network shown in Fig. 1(a) where all the edges have nit capacity. It can be verified that R 1 = R 2 =1is not feasible with roting, yet feasible with network coding as shown in Fig. 1(a). It is no longer clear how to interpret this code as concrrent flows. Note that the data transmitted on edge 4 is b 1 b 2 and knowledge of b 2 is needed at d 1 in order to recover b 1.We interpret the transmission of b 1 b 2 as a poisoning of the two flows and the transmission of b 2 (b 1 ) on edge 5 (3) as an antidote. We also introdce the notion of an antidote reqest which notifies node s 1 (s 2 ) of the poisoning downstream and reqests an antidote to d 2 (d 1 ). Given an arbitrary network and mltiple nicast connections, we wish to identify network codes sing this poison-antidote approach. In order to do so, we enrich the variable-space by introdcing variables p e (i j, ),q e (i j, ),r e (i j, ) for every vertex. These variables respectively keep track of the poison, antidote reqest, and antidote associated with the poisoning of ser j s data by ser i sdataatvertex. This is done to ensre that the poison originating at a particlar vertex is remedied by the appropriate antidote. In this section, we derive sfficient conditions for the existence of a network coding soltion involving roting, dplicating, and bit-level XOR. Theorem 1: For a network G =(V, E), with the capacity of edge e given by C e and the ranges of the variables given by v, V, e E, 1 i n, 1 j n, and i j, consider the following set of eqations. For all v,, i, and j: p e (i j, )+q e (i j, )+r e (i j, ) = p e (i j, )+q e (i j, )+r e (i j, ) (3) q e (i j, ) (b) r(2 1) = 1 d 1 { 0 if v = q e (i j, ) 0 otherwise (4) p e (i j, ) For all, i, and j: { 0 if v = p e (i j, ) 0 otherwise (5) p e (j i, ) =p e (i j, ) if e Γ O () (6) At every edge e: n max(p e (i j, ),p e (j i, )) + x e (i) i<j + r e (i j, ) z e C e (7) At every edge e and j: x e (j)+ (p e (i j, )+q e (j i, )) 0 (8) i x e ( ) 0, r e ( ) 0, p e ( ) 0, q e ( ) 0 If the conditions specified in (2) - (8) are satisfied, then the rate triples (s i,d i,r i ) for i =1, 2,...,nare feasible with network coding. Frther, the only operations involved are dplication, roting, and bit-level XOR. Proof: We assme that the variables satisfying (2) - (8) are integers, which is jstified by scaling the network over time [6]. We begin by making the following observations. Observation 1: From (4) we see that antidote reqest q e (i j, ) (a negative qantity) can terminate only at node. Observation 2: From (5) we see that poison p e (i j, ) (a negative qantity) can originate only at node. Observation 3: For edge e, let f e (i j, ) = p e (i j, ) +q e (i j, ) +r e (i j, ). Fix i, j, and. From (3) we see that f e (i j, ) is conserved at each node. In the example shown in Fig. 1(b), f e (1 2,) = 1 for e = 3 and f e (1 2,)= 1 for e =1, 4, 6. Notice that the edges 1,3,4, and 6 form a cycle if we reverse the direction of edges with f e (1 2,)= 1, i.e., 1,4, and 6. In general, for any network, f e (i j, ) can be decomposed as a sperposition of cycles where f e(i j, ) =±1 for all edges e with the nderstanding that if f e(i j, ) = 1, then the direction of e is reversed. De to pacity of space, we skip the proof here. We refer to each of these cycles as an f (i j, ) cycle for i, j, and. Frther, since the qantities p( ), q( ), and r( ) are integers, it follows that along any edge of an f (i j, ) cycle, only one of these qantities is nonzero. From Observations 1 and 2 we see that q e (i j, ) terminates only at and that p e (i j, ) originates only at. (To simplify notation we represent a path from 1 to v m as ( 1,v m ). The distinction between an edge (, v) and a path ( 1,v m ) is clear from the context.) Ths, any f (i j, ) cycle comprises three paths (v, ), (, w) and (v, w) sch that (v, ) satisfies q e (i j, ) = 1, (, w) satisfies p e (i j, ) = 1, and (v, w) satisfies r e (i j, ) = 1. Some f (i j, ) cycles might have v =, which wold mean that it comprises of only the poison and antidote paths. 1759

3 From (6), we see that if tail(e) = then we can associate p e (j i, ) with p e (i j, ). This can be extended and we can associate an f (j i, ) cycle with an f (i j, ) cycle. Ths, all the poison, antidote, and antidote-reqest variables can be written as a sperposition of pairs of f (i j, ) and f (j i, ) cycles. We show that we can perform network coding on each of these pairs of cycles. Consider one sch pair of cycles. Let the cycles be f (i j, 3 ) and f (j i, 3 ). From the discssion above, the f (i j, 3 ) cycle is composed of paths ( 1, 3 ) with q e (i j, 3 ) = 1, ( 3, 4 ) with p e (i j, 3 ) = 1, and ( 1, 4 ) with r e (i j, 3 ) = 1. Similarly, the f (j i, 3 ) cycle is composed of paths ( 2, 3 ) with q e (j i, 3 )= 1, ( 3, 5 ) with p e (j i, 3 )= 1, and ( 2, 5 ) with r e (j i, 3 )=1. Using (8), we note that we can associate a niqe nit-rate path, say, x e(i) =1to the paths ( 1, 3 ) and ( 3, 5 ). Similarly we can associate a niqe nit-rate path say x e(j) =1to the paths ( 2, 3 ) and ( 3, 4 ). In the example shown in Fig. 1(b), we have 1 = s 1, 2 = s 2, 3 =, 4 = d 1, and 5 = d 2. Note that x e (1) = 1 (x e (2) = 1)) fore =1, 4, 7. (e =2, 4, 6.) Let b 1 represent a nit rate bit stream corresponding to x(i) available at 1 and b 2 represent a nit rate bit stream corresponding to x(j) available at 2. We perform the following network coding in order to reliably transmit one nit data from 1 to 5 and one nit of data from 2 to 4. b 1 is transmitted along the paths ( 1, 3 ) and ( 1, 4 ). b 2 is transmitted along the paths ( 2, 3 ) and ( 2, 5 ). b 1 b 2 is transmitted along the paths ( 3, 4 ) and ( 3, 5 ). This encoding, in effect, is eqivalent to replacing x e(i),x e(j) and the f (i j, 3 ),f (j i, 3 ) cycles by two nit-capacity virtal edges e 1 =( 1, 5 ) and e 2 =( 2, 4 ) sch that x e1 (i) =1and x e2 (j) =1. This replacement ensres that the nmber of f (i j, )and f (j i, ) cycles is redced while still ensring that x e (i) and x e (j) (the affected flows) satisfy the flow-conservation eqations. For this encoding, it can be verified that the sage of all edges along the paths ( 1, 3 ), ( 2, 3 ), ( 3, 4 ), ( 3, 5 ), ( 1, 4 ), and ( 2, 5 ) is given by the expression max(p e (i j, 3 ),p e (j i, 3 )) + x e (i)+x e (j) +r e (i j, 3 )+r e (j i, 3 ) (9) which evalates to one along all the edges. Ths, the sage of the edges in this pair of f (i j, 3 ) and f (j i, 3 ) cycles is given by (9). Note that the terms r e (i j, 3 )+r e (j i, 3 ) correspond to physical transmissions of bits. The term x e (i)+x e (j) can be thoght of as the data that wold be transmitted if data is sent by roting. However, by sending b 1 b 2, we transmit at a lower rate, lower by max(p e (i j, 3 ),p e (j i, 3 )). These can be thoght of as savings obtained de to network coding. Using Observation 3, we see that f e (i j, ) is a sperposition of pairs of cycles, each of which can be replaced by a pair of virtal edges over which roting sffices. By performing this operation till there are no more sch cycles left, we end p with an eqivalent network over which the connection triples (s i,d i,r i ) for i =1, 2,...,nare feasible with roting. If all the data is transmitted by roting, then the load on any edge is n x e(i). However, recall that de to an f (i j, ) cycle, the load on the edges in the cycle is adjsted by the qantity in (9). Since f e (i j, ) is a sperposition of many f (i j, ) cycles, the net adjstment of the load on the edges is obtained by smming (9) over all the f (i j, ) cycles. This expression is given by the left hand side of (7). The interpretation of the varios terms appearing in the smmation is as follows: By network coding, we obtain a savings of max(p e (i j, ),p e (j i, )) (which is negative) which is offset by excess load of r e (i j, ) (which is positive) de to the remedies. Usally, the savings are non-zero on congested edges and the remedies are non-zero on edges with spare capacity, ths allowing s to trade off bottleneck links with excess capacities. Every soltion that satisfies the constraints (2) - (8) corresponds to a network code that transmits data at a rate z e (see (7)) on edge e. These constraints can be combined with several possible objective fnctions, e.g., one can maximize the total throghpt (the sm of rates i R i) or minimize the cost of commnicating certain fixed rates sbject to the capacity constraints of the graph. III. INTEGER PROGRAMMING FORMULATION The previos linear program offers some flexibility in the choice of objective fnctions, permits roting (and network coding) along several paths from sorce to destination and makes little assmption abot the nderlying network topology. However, it reslts in large optimization problems, both in terms of the nmber of constraints and the nmber of variables. We will discss the complexity in Section IV. In particlar, keeping track of the origin of the poison increases the complexity drastically. (The term in p e (i j, ) can be thoght of as the variable keeping track of the origin of the poison.) This motivates s to consider a simpler optimization problem, where we restrict the connections to be of nit rate. 1 This makes the reslting optimization problem an integer program, bt with fewer constraints and variables than the original formlation. Consider the following linear program, where throghot 1 i, j n, and i j: at every node v V: x e (i) r e (i) p e (i, j) min e E a e z e +1 if v = s i x e (i) = 1 if v = d i 0 otherwise r e (i) p e (i, j) (x e (i)+ (10) n p e (i, j)) j=1 (11) r e (j) (12) 1 Higher rates can be modeled by placing several connections between the same sorce-destination pair. 1760

4 i<j at every destination node d i : at every edge e E: e Γ O(d i) j=1 n p e (i, j) =0 (13) n n max(p e (i, j),p e (j, i)) + x e (i)+ r e (i) z e C e x e (i)+ (14) n p e (i, j) 0 (15) j=1 x e ( ) 0, r e ( ) 0, p e ( ) 0 Note that we reqire flows of nit rate (10), ths losing some flexibility in the problems that can be modeled. The cost fnction takes the form of a minimm cost flow problem, with cost coefficients a e mltiplying the sage z e of every edge. Frthermore, we reqire the nderlying graph to be acyclic, which becomes necessary to exclde invalid network coding soltions. In the following we prove that soltions of this integer program are in fact valid network codes, which can be constrcted by sing roting and the binary XOR operation. Lemma 1: The cost of the optimal soltion of the linear program (10)-(15) is smaller or eqal to the optimal cost of the associated mlti-commodity flow problem (10), (16), and (17). Proof: Set all variables r e ( ) and p e ( ) to zero. Then, the linear program redces to the following form: min e E a e z e sbject to (10) at every node v Vand n x e (i) z e C e (16) x e (i) 0 (17) at every edge e E. This is identified to be the associated mlti-commodity flow problem consisting of the flow conservation constraints at the nodes, the constraint that the edge sages z e do not exceed the edge capacities C e and the nonnegativity constraints on the flows. The optimal soltion to this mlticommodity flow problem is a basic feasible soltion to the linear program (10)-(15) and therefore is also an pper bond on its optimal soltion. Contining along similar lines, it is possible to prove that the optimal cost of the linear program (2)-(8) is also lower than the cost of the optimal soltion of the associated mlti-commodity flow problem Theorem 2: If a soltion to the linear program (10)-(15) is integer, then it corresponds to a feasible network coding soltion involving only the binary XOR as a coding operation. Proof: Assme, we have an integer soltion. Since the flows are of nit rate we have x e (i) {0, +1}. Frthermore, from (15) we conclde that p e (i, j) {0, 1}. If all p e (, ) variables are zero, then according to the previos lemma the Variables Constraints Linear Program 3 E V n(n 1) V (n 2 n)(3 V + E ) Integer Program E n(n 1) (n 2 + n) V +(n +1) E TABLE I THE COMPLEXITY OF BOTH OPTIMIZATION PROBLEMS. WE APPROXIMATE THE NUMBERS UP TO LEADING TERMS. soltion corresponds to a mlti-commodity flow problem and therefore the connections can be established by roting only. Assme now some p(, ) variables eqal to 1 and pick an edge e (with tail(e) = and head(e) =v), where p e (i, j) = 1. (15) implies that x e (i) =1on this edge also. Since the flows are of rate one and we insist on integer soltions, flow i takes precisely one path from its sorce s i to its destination d i and clearly edge e lies on this path. The poison p e (i, j) satisfies a conservation law of the form (12), and therefore forms a path which can se only edges on the path of flow i (15). From (13) the p e (i, j)-path has to terminate at a node w d i 2. At the termination point w of the p e (i, j)-path an antidote r(j) has to enter the node (12). This antidote also forms a path (11) and let its sorce be node t. The RHS of (11) has to be strictly positive at node t, which implies that the flow x(j) entering node t mst be one, and the poison p(j, ) entering t mst be zero. Given a soltion to the proposed integer linear program, we constrct the network code by transmitting the following bit streams on edge e: x(i) if x e (i) =1and p e (i, ) =0, x(i) x(j) if x e (i) =1and p e (i, j) = 1 for some j, x(i) if r e (i) =1. We have proved, that every poison is canceled by the appropriate antidote, and that every antidote has its origin at a node, where npoisoned flow of the proper type enters. The assmption that the graph is acyclic is needed to ensre that the antidote comes from a node pstream of the poisoning node. This shows the validity of the proposed code constrction scheme and completes the proof. IV. PERFORMANCE EVALUATION Conting the nmber of variables and constraints (see Table I) of both formlations, we see that the integer program is of considerably smaller size than the linear program. On the other hand, solving an integer program is in general more complex than solving a linear program withot integer constraints. Integer programming is NP-complete [8] and networks exist where the integer program is very difficlt to solve. To find integer soltions we have sed a linear programming-based branch-andbond algorithm. In general one has to consider the trade-off between the redced size of the optimization problem (10)-(15) and the increase in complexity de to branch-and-bond. The first setp that we considered for performance evalation of or network coding scheme is the freqently sed geometric random graph model. Here, a nmber of nodes are scattered niformly over a nit sqare and nodes whose distance is below a certain threshold are considered connected with a point-to-point link. We have condcted extensive simlations 2 Since we reqire a directed acyclic graph, the vertices can be topologically sorted, thereby defining an order on the vertex set V. 1761

5 on random geometric networks, sing different objective fnctions sch as total throghpt or minimm cost, bt cold not observe gains. This observation motivates s to consider more reglar grid networks, sch as the one depicted in Fig. 2. We illstrate the performance of the integer program 3 on the grid network as docmented in Table II. We consider no capacity constraints and a minimm-cost objective fnction. The random parameters in the simlations are the cost coefficients of the links. We consider two different distribtions according to which they are drawn. The first distribtion models a scenario where most links have low cost, and a few links have large cost, ths representing bottlenecks. In this setp we observe gains de to network coding. When the link costs are niformly distribted, however, the gains disappear. A comparison with the simlation reslts given by the athors in [9] and more recently [7] is in order. In particlar, in [7] the athors consider a wireless network, which ses geographic (min-hop) roting. They report large gains in the total throghpt, when implementing an opportnistic heristic code constrction techniqe. These gains can be explained with several factors, the most important being the lack of congestion control when sing solely geographic roting. If the rotes are chosen based on geography, withot taking congestion into accont, a drastic breakdown of the total throghpt is experienced as the nmber of flows is increased. Network coding acts here as a congestion resoltion techniqe and gives large gains. Another fndamental difference between the setp in [7] and or model is that the former works on a wireless network, where nodes broadcast their messages to a set of neighbors, a phenomenon referred to as the wireless mlticast advantage [10]; we on the other hand consider a wired network with point-to-point links. The gains that we report are fairly modest, compared to [7,9]. However, these reslts hold only for the narrow setp that we consider, namely wireline networks, optimal roting 4 as the benchmark and no consideration of practical implementation isses sch as protocol design. Frthermore, we assme the flows in the network static and do not consider the dynamic behavior of the network. It is worthwhile to investigate the tility of network coding, when any of these restrictions is relaxed. Based on the previos discssion, larger gains might be observed in wireless networks (de to the wireless mlticast advantage), when the cost coefficients or capacities of the links are distribted highly non-niform, or when considering network coding as a tool to enhance roting, e.g. as a congestion resoltion techniqe. Also, frther research is indicated on the dynamic aspects of networks sing network coding and the interplay between practical network design and network coding. V. CONCLUSIONS Few explicit code constrction techniqes are known for the mltiple nicast setp. In this paper we proposed a systematic approach to introdce sboptimal, yet practical network coding for mltiple nicasts. Or scheme comes in two variations, a linear program and an integer program of smaller size. We discssed the comptational complexity and based on simlations a b c d e f Fig. 2. The grid network sed for simlations. Note, that all edges are directed from the sorces a, b, c down to the sinks d, e, f. Approach Connections Networks Average with gains gain a i 1 wp % 28.0% 100 wp % 24.9% a i nif(1, 100) TABLE II SIMULATION RESULTS ON THE GRID NETWORK IN FIG. 2:ANUMBER OF CONNECTIONS IS RANDOMLY SELECTED FROM THE DESIGNATED SOURCE AND DESTINATION PAIRS. THE LINK COSTS ARE RANDOMLY FIXED ACCORDING TO THE SPECIFIED DISTRIBUTION. WE REPORT THE FRACTION OF NETWORKS, WHERE NETWORK CODING RESULTED IN A LOWER TRANSMISSION COST THAN ROUTING. THE COST GAIN IS DEFINED AS cost roting cost coding AND IS CALCULATED OVER THE NETWORKS, cost roting WHERE A GAIN WAS OBSERVED. evalated the network coding gains. We also compared them with recently pblished reslts by other athors. REFERENCES [1] R. Ahlswede, N. Cai, S.-Y. R. Li, and R. W. Yeng, Network information flow, IEEE Trans. on Inf. Theory, vol. IT-46, pp , [2] T. Ho, M. Médard, R. Koetter, D.R. Karger, M. Effros, J. Shi, and B. Leong, A random linear network coding approach to mlticast, sbmitted to IEEE Trans. on Inf. Theory. [3] D. S. Ln, M. Medard, T. Ho, and R. Koetter, Network coding with a cost criterion, in Proc. Intl. Symp. on Inf. Theory and its Appl., [4] D. P. Berstekas, Network optimization: Continos and discrete models, Belmont, MA: Athena Scientific, [5] N. Ratnakar, D. Traskov, and R. Koetter, Approaches to network coding for mltiple nicast, in Intern. Zrich Seminar on Comm., Febr [6] R. Koetter and M. Médard, An algebraic approach to network coding, IEEE/ACM Trans. on Networking, vol. 11, no. 5, pp , Oct [7] S. Katti, D. Katabi, Wenjn H, and Rahl Hariharan, The importance of being opportnistic: Practical network coding for wireless environments, in Allerton Conference, [8] I. Barland, P.G. Kolaitis, and M.N. Thakr, Integer programming as a framework for optimization and approximability, In Proc. IEEE Conference on Comptational Complexity, pages , [9] R. Khalili and K. Salamatian, A new relaying scheme for cheap wireless relay nodes in Wiopt 2005, Trentino, Italy. [10] D.S. Ln, N. Ratnakar, M. Medard, R. Koetter, D.R. Karger, T. Ho, E. Ahmed, Minimm-cost mlticast over coded packet networks, sbmitted to IEEE Trans. on Inf. Theory. 3 Since both the linear and the integer program model the same problem, the linear program in this case wold have given eqivalent reslts. 4 By optimal roting we mean the optimal soltion of the associated mlticommodity flow problem. 1762