Sum Degrees-of-Freedom of Two-Unicast Wireless ... - CiteSeerX

Sum Degrees-of-Freedom of Two-Unicast Wireless Networks Ilan Shomorony and A. Salman Avestimehr School of Electrical and Computer Engineering, Cornell University, Ithaca, NY, USA [email protected] [email protected] Abstract—We consider two-source two-destination (i.e., twounicast) multi-hop wireless networks that have a layered structure with arbitrary connectivity. We show that, if the channel gains are independently drawn from continuous distributions, then, with probability 1, two-unicast layered Gaussian networks can only have 1, 3/2 or 2 sum degrees-of-freedom1 . We provide necessary and sufficient conditions for each case based on the network topology and a new notion of source-destination paths with manageable interference.

I. I NTRODUCTION Characterizing network capacity is one of the central problems in network information theory. While this problem is in general unsolved, there has been considerable success in two research fronts. The first one focuses on single-flow multihop networks, in which one source sends the same message to one or more destinations. In this scenario, all destination nodes require the same message, and there is effectively only one information stream in the network. Starting from the max-flow min-cut theorem of Ford-Fulkerson [1], there has been significant progress on this problem. For wireline networks, the maximum multicast flow was characterized in [2] using random coding and in [3, 4] using linear network coding. In [5], the max-flow min-cut theorem was generalized for a class of linear deterministic networks with broadcast and interference. Inspired by this generalization, the multicast capacity of wireless networks was then characterized to within a gap that does not depend on the channel gains ([5]). The second research direction focuses on single-hop multiflow wireless networks, i.e., the interference channel (IFC). While the capacity of the IFC remains unknown (except in special cases, such as [6–9]), several approximations have been derived, such as constant-gap capacity approximations [10, 11] and degrees-of-freedom (DoF) characterizations ([12–14]). However, once we go beyond single-hop, much less is known about the capacity of multi-flow networks. Even for two-source two-destination networks there are few general results, such as [15], where the maximum flow in twounicast undirected wireline networks is characterized. For twounicast directed wireline networks, [16–18] provided graphtheoretic conditions under which rate (1, 1) can be achieved. In the wireless realm, constant-gap capacity approximations for certain two-hop networks were obtained in [19]. In [20], it 1 Unless the source-destination pairs are disconnected, in which case no degrees-of-freedom can be achieved.

was shown that the network resulting from the concatenation of two or more fully connected IFCs admits two DoF. In this paper, we consider two-unicast multi-hop wireless networks that have a layered structure with arbitrary connectivity. We consider an AWGN channel model and assume that the channel gains are independently drawn from continuous distributions and remain fixed during the course of communication. Moreover, we assume that all channel gains are known at all nodes. Under these assumptions, we show that, with probability 1 over the choice of the channel gains, two-unicast layered Gaussian networks can only have 1, 3/2 or 2 sum DoF. Moreover, we provide necessary and sufficient conditions for each case that are based only on properties of the network graph. We state our main result in Section II and describe its proof in Section III. Due to space limitations, we omit proof details and refer to [21] for complete proofs. II. D EFINITIONS AND M AIN R ESULT A multiple-unicast Gaussian network N = (G, L) consists of a directed graph G = (V, E), where V is the node set and E ⇢ V ⇥V is the edge set, and a set of source-destination pairs L ⇢ V ⇥ V . We consider two-unicast Gaussian networks, i.e., L = {(s1 , d1 ), (s2 , d2 )}, for distinct s1 , s2 , d1 , d2 2 V . We will assume that the network is layered, i.e., the node set V can be partitioned into Srr 1subsets V1 , V2 , ..., Vr (the layers) in such a way that E ⇢ i=1 Vi ⇥Vi+1 , and V1 = {s1 , s2 }, Vr = {d1 , d2 }. For v 2 Vj , we let I(v) , {u 2 Vj 1 : (u, v) 2 E} and O(v) , {u 2 Vj+1 : (v, u) 2 E}. Furthermore, we let `(v) be the index of the layer containing v, i.e., v 2 V`(v) . Notice that the layers induce a natural ordering of the nodes. For each edge e = (vi , vj ), we associate a real-valued channel gain he (or simply hi,j ). We will assume that the he ’s are independently drawn from continuous distributions and are fixed during the course of communication. We also assume that all he ’s are fully known at all nodes. At time m, each vi 2 V \ {d1 , d2 } transmits a real signal Xvi [m] (or Xi [m]), which must satisfy an average power constraint P . The signal received by vj 2 V \ {s1 , s2 } at time m is X Yj [m] = hi,j Xi [m] + Nj [m], for m = 1, 2, ... , vi 2I(vj )

where Nj [m] is the zero-mean unit-variance Gaussian discretetime noise process at vj . The transmitted signal from vj 2 V \ {s1 , s2 } at time m must be a function of its past received signals Yj [k], for k = 1, ..., m 1. Source si has a message Wi

that it wishes to send to di , and encodes it into transmit signals Xsi [m], m = 1, ..., n, for i = 1, 2, for a communication session of duration n. We say that rates Ri , log n|Wi | for i = 1, 2 are achievable if the probability of error in the decoding of both messages by their destinations can be made arbitrarily small by choosing a sufficiently large n. The sumcapacity C⌃ (P ) is the supremum of the achievable sum-rates. Definition 1. The sum degrees-of-freedom D⌃ is defined as D⌃ , lim

C⌃ (P ) . log P

P !1 1 2

Definition 2. A path Pv1 ,vk from v1 to vk is an ordered set {v1 , v2 , ..., vk } ⇢ V such that (vi , vi+1 ) 2 E for i = 1, ..., k 1. We write v1 ; vk , if there is a path Pv1 ,vk . Definition 3. Paths Pva ,vb and Pvc ,vd are said to be disjoint if Pva ,vb \ Pvc ,vd = ;. Definition 4. For S ⇢ V , we say that G[S] is the graph induced by S on G, if G[S] = (S, Es ), where Es = {(vi , vj ) 2 E : vi , vj 2 S}.

Definition 5. N 0 = (G0 , L) is a subnetwork of N = (G, L), if G0 = G[S], for some S ⇢ V such that L ⇢ S ⇥ S.

Next, we assume that we have two disjoint paths Ps1 ,d1 and Ps2 ,d2 . We let ¯i = 2 if i = 1 and ¯i = 1 if i = 2. Definition 6. We say that va 2 / Psi ,di causes interference on I Psi ,di and write va ; Psi ,di if we can find vb 2 Psi ,di and a path Ps¯i ,va such that (va , vb ) 2 E and Ps¯i ,va \ Psi ,di = ;, for I i = 1 or 2. We write va ! Psi ,di , if, in addition, va 2 Ps¯i ,d¯i .

Consider a subnetwork (G[S], {(s1 , d1 ), (s2 , d2 )}) for S I Ps1 ,d1 [ Ps2 ,d2 . Let ni (G[S], Psi ,di ) , |{v 2 S : v ; Psi ,di }| I and nD i (Ps¯i ,d¯i , Psi ,di ) , |{v 2 V : v ! Psi ,di }|, for i = 1, 2. I Notice that the path implied by v ; Psi ,di must exist in the subnetwork. If there is no ambiguity in the choice of Ps1 ,d1 and Ps2 ,d2 , we will simply use ni (G[S]) and nD i . Definition 7. Two disjoint paths Ps1 ,d1 and Ps2 ,d2 have manageable interference if we can find S ⇢ V so that Ps1 ,d1 , Ps2 ,d2 ⇢ S, n1 (G[S]) 6= 1 and n2 (G[S]) 6= 1.

Theorem 1. For a two-unicast layered Gaussian network N = (G = (V, E), {(s1 , d1 ), (s2 , d2 )}) where the channel gains are chosen according to independent continuous distributions, with probability 1, D⌃ is given by A) 1 if N contains a node v whose removal disconnects di from {s1 , s2 } and s¯i from {d1 , d2 }, for i = 1 or 2, A0 ) 1 if N contains an edge (v2 , v1 ) such that the removal of v1 disconnects di from {s1 , s2 } and the removal of v2 disconnects s¯i from {d1 , d2 }, for i = 1 or 2, B) 2 if N contains two disjoint paths Ps1 ,d1 and Ps2 ,d2 with manageable interference (see Definition 7), B0 ) 2 if N or any subnetwork does not contain two disjoint paths Ps1 ,d1 and Ps2 ,d2 , but is not in case (A), C) 32 in all other cases.

III. P ROOF OVERVIEW In this section, we describe the basic ideas in the proof of Theorem 1. For detailed proofs, we refer to [21]. We consider cases (A), (A0 ), (B), (B0 ) and (C) sequentially. The intuition behind (A) is as follows. Let W1 be the message from s1 and W2 be the message from s2 . If the removal of v disconnects d1 from both sources, then by knowing the received signal at v we should be able to decode W1 . Then, since v also disconnects d2 from s2 , loosely speaking, all the information about W2 goes through v. Therefore, v can use the knowledge about W1 to remove any interference due to signals about W1 , thus being able to decode W2 as well. Since a single node can decode both messages, we have that D⌃  1, and it follows that D⌃ = 1, since 1 degree-offreedom is trivially achievable from the fact that s1 ; d1 and s2 ; d2 . The intuition behind (A0 ) is similar. If the removal

s1

d1

s2

d2

Fig. 1.

An example of a network in case (A0 ).

of v1 disconnects d1 from both sources, then by knowing the received signal at v1 we should be able to decode W1 . Since the removal of v2 disconnects s2 from both terminals, all the information regarding W2 goes through v2 . This means that all the information received at v1 which does not come from v2 is about W1 and, thus, by knowing the received signal at v1 , one can remove the part regarding W1 and obtain the part of the transmitted signal at v2 regarding W2 . But this implies that from v1 we should be able to decode both W1 and W2 , which implies D⌃  1. An example of a network that would fall in (A0 ) is shown in Figure 1. A. Networks with two degrees-of-freedom Since the cut-set bound trivially yields D⌃  2, for cases (B) and (B0 ), we only need to describe achievability schemes for 2 degrees-of-freedom. In both cases, we will start by identifying key layers, whose nodes will perform non-trivial relaying operations. All nodes which do not belong to key layers will forward their received signal. This allows us to build a condensed network Nc , which only contains V1 , Vr and the key layers. The connectivity and channel gains are determined according to the effective transfer matrices between consecutive layers of Nc . An example is shown in Figure 2. We refer to the channel gain of edge (v, u) from Nc ˆ u). by h(v, In order to achieve two degrees-of-freedom, we consider two types of transmission strategies, according to the structure of the condensed network. If the condensed network is a 2 ⇥ 2⇥2 interference channel, we use the scheme described in [20]

v1 s1

v2

d1

v3 s2

d2

vm

(a) Fig. 2.

(b)

A 5-layer network (a) and its 3-layer condensed version (b)

to achieve D⌃ = 2. Otherwise, we describe an amplify-andforward scheme that guarantees that the end-to-end transfer matrix for the network (and also for the original h condensed i 1 0 network) is 0 2 , for 1 , 2 6= 0. Thus we have Ydi = eff eff i Xsi + Ndi , for i = 1, 2, where Ndi is the effective additive noise at di , and we have two AWGN channels. We restrict the sources to using power ↵P , for ↵ 2 (0, 1). It can be seen that, for P sufficiently large, ↵ can be chosen independent of P so that the transmit power constraint is simultaneously satisfied at all relays. Since the scaling factors used at the key layers and the variance of Ndeffi , i2 , are functions of the channel gains only (and not P ), source-destination pair⌘(si , di ), for i = 1, 2, ⇣ ↵ i2 P 1 can achieve rate Ri = 2 log 1 + 2 , and, therefore, we i achieve D⌃ = 2. First, we consider (B). Thus we have disjoint paths Ps1 ,d1 and Ps2 ,d2 with manageable interference, i.e., 9S ⇢ V such that Ps1 ,d1 [ Ps2 ,d2 ⇢ S, n1 (G[S]) 6= 1 and n2 (G[S]) 6= 1. We assume that S is minimal, and that all nodes in V \ S are removed. If n1 (G[S]) = 0 and n2 (G[S]) = 0, then achieving D⌃ = 2 is trivial, since there is no interference. If ni (G[S]) 2, for i = 1 or 2, we let vpi be the first node in Psi ,di whose removal disconnects di and s¯i . If ni (G[S]) 2, V`(vpi ) 1 is the last layer where we can choose the scaling factors to cancel the interference on Psi ,di , and it will be a key layer. In the following Lemmas, it is assumed that ni (G[S]) 2, for i = 1 or 2, and thus vpi is defined. The proofs are in [21]. Lemma 1. There exist two paths Ps1 ,vpi and Ps2 ,vpi such that Ps1 ,vpi \ Ps2 ,vpi = {vpi }.

Fig. 3. Illustration of a condensed network with n1 (G[S]) and n2 (G[S]) = 0. Solid lines represent edges that must exist in the condensed network, while the dashed lines represent edges that may not exist.

ˆ 1 , vi )h(v ˆ i , d2 ) = 0 for i = 1, ..., m and T2,1 we must have h(s (the bottom left entry in T ) is always 0. From Lemma 1, we can find two nodes va , vb 2 I(vp1 ) ⇢ V`(vp1 ) 1 with associated variables xa and xb , and two disjoint paths Ps1 ,va and Ps2 ,vb . From Lemma 2, we can find vc 2 I(vp1 ) ⇢ V`(vp1 ) 1 , such that s2 ; vc and c 6= m. We now claim that if the matrices  ˆ , v )h(v ˆ , d ) h(s ˆ , v )h(v ˆ ,d ) h(s M1 = ˆ 1 a ˆ a 1 ˆ 1 b ˆ b 1 and h(s2 , va )h(va , d1 ) h(s2 , vb )h(vb , d1 )  ˆ , v )h(v ˆ , d ) h(s ˆ , v )h(v ˆ h(s ,d ) M2 = ˆ 2 c ˆ c 1 ˆ 2 m ˆ m 1 h(s2 , vc )h(vc , d2 ) h(s2 , vm )h(vm , d2 ) are both full-rank, then we can choose x1 , ..., xm so that T is as desired. To see this, consider x0 = [x01 ... x0m ], where x0j = 0 for j 6= a, b, and [x0a x0b ]T = M1 1 [1 0]T . This choice of scaling factors results in T1,1 = 1 and T1,2 = 0. If T2,2 6= 0 we are done. Otherwise, if T2,2 = 0, we let x00 = [x001 ... x00m ], where x00j = 0 for j 6= c, m and [x00c x00m ]T = M2 1 [0 1]T . This choice results in T1,2 = 0 and T2,2 = 1. If we have T1,1 6= 0, we are done. Otherwise, we set x000 = x0 + x00 . By linearity, this choice will guarantee that T is the identity matrix. Next we show that, with probability 1 over the choice of the he ’s, M1 and M2 are full-rank. First we consider the transfer matrix between (s1 , s2 ) and (va , vb ), given by  ˆ , v ) h(s ˆ ,v ) h(s Z1 = ˆ 1 a ˆ 2 a . h(s1 , vb ) h(s2 , vb )

The determinant of Z1 can be seen as a polynomial in the channel gains he . If det Z1 is not identically zero, since the i Lemma 2. There are (at least) two nodes v1 , v2 2 I(vp ) such he ’s are drawn independently from continuous distributions, det Z1 will be non-zero w.p. 1. To see that det Z1 is not that s¯i ; v1 and s¯i ; v2 . identically zero, notice that the existence of disjoint paths We describe the achievability scheme when n1 (G[S]) 2 Ps1 ,va and Ps2 ,vb guarantees that, if we set he = 1 if e and n2 (G[S]) = 0, and thus only vp1 is defined. The case connects adjacent nodes of Ps1 ,va or Ps2 ,vb and he = 0 where n1 (G[S]) 2 and n2 (G[S]) 2 is considered in [21]. otherwise, Z1 will be the identity matrix. Therefore, Z1 will Our condensed network is formed by layers V1 , V`(vp1 ) 1 and be invertible, and det Z1 cannot be identically zero. Now, ˆ a , d1 )h(v ˆ b , d1 ) det Z1 . Since Vr , with m = |V`(vp1 ) 1 |, as shown in Figure 3. To each vi 2 we notice that det M1 = h(v ˆ a , d1 )h(v ˆ b , d1 ) is also V`(vp1 ) 1 , i = 1, ..., m , we associate a scaling factor xi . We va ; d1 and vb ; d1 , we have that h(v must show that the end-to-end transfer matrix, given by a non-identically zero polynomial in the he ’s, and therefore Pm M1 is invertible w.p. 1. To show that M2 is invertible w.p. ˆ 1 , vi )h(v ˆ i , d1 )xi Pm h(s ˆ 2 , vi )h(v ˆ i , d1 )xi h(s i=1 P T = Pi=1 , 1, we follow very similar steps, by noticing that the transfer m ˆ m ˆ ˆ ˆ i=1 h(s1 , vi )h(vi , d2 )xi i=1 h(s2 , vi )h(vi , d2 )xi matrix between {vc , vm } and {d1 , d2 } is full-rank w.p. 1, since can be made diagonal with non-zero diagonal entries by we have disjoint Pvc ,d1 and Pvm ,d2 . an appropriate choice of x1 , ..., xm . Since, in this case, The proof for n1 (G[S]) 2, n2 (G[S]) 2 follows similar n2 (G[S]) = 0, there is no path from s1 to d2 , and therefore steps, except if our condensed network is a 2 ⇥ 2 ⇥ 2 inter-

ference channel, in which case we apply the real interference alignment scheme described in [20]. If our network N is in case (B0 ), we proceed as follows. We use a result provided in [17, 18] to claim that if N is not in case (A), then it contains two disjoint paths Ps1 ,d1 and Ps2 ,d2 , a Butterfly subnetwork or a Grail subnetwork. Since we have a subnetwork with no two disjoint paths which is not in case (A), we must have a Butterfly or a Grail network. In each case we identify key layers and provide schemes as the one above to achieve D⌃ = 2. See [21] for details. B. Networks with 3/2 degrees-of-freedom For case (C), we only need to consider networks which have two disjoint paths Ps1 ,d1 and Ps2 ,d2 , but do not have two disjoint paths with manageable interference. This is because all networks which do not contain two disjoint paths Ps1 ,d1 and Ps2 ,d2 must fall into (A) or (B0 ). Moreover, any network that has two disjoint paths with manageable inteference will fall into (B). It can then be shown that our network may be assumed WLOG to be in one of two cases: D C1. n1 (G) 2, nD 1 = 1, n2 (G) = 1 and n2 = 0. D C2. n1 (G) = n1 = 1 Consider a network that falls in case C1. Notice that we must I have a node v1 2 / Ps1 ,d1 [ Ps2 ,d2 such that v1 ; Ps1 ,d1 and thus we have a path Ps2 ,v1 disjoint from Ps1 ,d1 . We let v2 be the last node in Ps2 ,d2 \ Ps2 ,v1 , and we have a path Pv2 ,v1 . If we let S ⇤ = Ps1 ,d1 [ Ps2 ,d2 [ Pv2 ,v1 , we have n1 (G[S ⇤ ]) 2. Since Ps1 ,d1 and Ps2 ,d2 do not have manageable interference, we must have n2 (G[S ⇤ ]) = 1. Since nD 2 = 0, we conclude I that we must have v3 2 Pv2 ,v1 \ {v2 } such that v3 ; Ps2 ,d2 , ⇤ and we must have a path Ps1 ,v3 ⇢ S . One example of such a network can be seen in Figure 4.

s1

v7

v8

v9 v10

s2

v6 Fig. 4.

v2

d1 v11 v3

v1

v5

v4 v

d2

Example of the network in case C1.

To achieve 3/2 DoF, we use a scheme based on two distinct modes of operation for the network, as illustrated in Figure 5. During Mode 1, we let the unique node in Ps2 ,d2 \ V`(v3 ) function as a destination d02 . Notice that we have disjoint paths Ps1 ,d1 and Ps2 ,d02 with manageable interference. In Mode 2, d02 becomes a source s02 , and we again have disjoint paths Ps1 ,d1 and Ps02 ,d2 with manageable interference. Therefore, if d02 = s02 stores the received signals during Mode 1, and forwards them during Mode 2, we can achieve 32 DoF. See [21] for details. A complete discussion of the converse can be found in [21]. Here, we will provide an intuition for the converse inequalities for the network in Figure 4. We assume that for each P and n we have a coding scheme for the network with node power constraint P and blocklength n. Then we let W1 and W2 be

X1(1)#

S1#

S2#

D2# D’# X2(1)#

D1#

S1#

D2#

S2#

X1(2)# D1# S2# D’# X2(1)#

D2#

Fig. 5. Depiction of Modes 1 and 2 for the achievability scheme in case C1.

chosen independently and uniformly at random among the possible messages, which induces random variables Xv and Yv for each node v. We will let A = {v 2 V : s2 6; v} and B = {v 2 V : s1 6; v}. To derive the converse inequalities, we consider a decomposition of the unit-variance Gaussian noise Nj at each node vj into m independent components with variance 1/m, where m = |I(vj )|. We associate each component with an incoming edge, and we define, for vi 2 I(vj ), ˜ i,j , hi,j Xi + Ni,j , X

where Ni,j is the noise P term associated with the edge (vi , vj ). Notice that Nj = i:vi 2I(vj ) Ni,j . We can now write, for a P ˜ i,j . We also define node vj , Yj = i:vi 2I(vj ) X ˜ i , {X ˜ i,j : j s.t. vj 2 O(vi )}. X

We let XS be the set of all Xi ’s , for vi 2 S, and Xin be a length n vector whose entries are the Xi [m]’s , for m = ˜S , X ˜ n , Y n and N n . 1, ..., n. Analogous definitions hold for X i j j In deriving the upper bounds to the rates, it is important to notice the “bottleneck” effect of Z structures. If we have a Z structure across two layers in the network, as shown in ˜ n and Y n , one can subtract X ˜n Figure 6(a), then, given X a b a,b ˜ n . Therefore, “almost all” information from Ybn and obtain X c,b ˜ n can be deduced from (Y n , X ˜ n ), and the conditional in X c a b n ˜n n ˜n mutual information I(Xc ; Xc |Yb , Xa ) can be upper bounded by a constant that does not depend on P . The next Lemma generalizes this notion to the structure in Figure 6(b).

va

vb

A

vb

vc

vd

vc

D

(a) Fig. 6.

(b) The Z structure.

Lemma 3. Suppose we have nodes vb and vc such that (vc , vb ) 2 E. Suppose, in addition, that we have sets A, S ⇢ V such that I(vb ) \ {vc } ⇢ A and for u 2 S [ A, u 6= vc , we have vc 6; u. Then, we have n ˜ cn |Ybn , X ˜A I(XSn ; X )  nK,

where K is only a function of the he ’s and the network N .

In order to provide an intuition of why D⌃  3/2, we consider, for a given n, the quantities n ˜n ˜n; X ˜ n) I(X9n , X I(X9n ; Y11 | XB ) 3 B ↵ , lim inf and , lim inf . n n P !1 P !1 log P log P 2 2 It is easy to see that 0  ↵,  1. Intuitively, ↵ can be thought of as the fraction of the transmit signal of v3 that is used to send useful information (i.e., information from v9 or B), and as the fraction of the transmit signal of v9 that is used to send information about W1 . Next we will state the converse inequalities and provide intuitive explanations. These inequalities are simple consequences of inequalities that are formally derived in [21]. We let Di be the degrees-of-freedom assigned to (si , di ), for i = 1, 2. Since all information from W1 must flow through v9 , we have D1  .

(1)

D1 + D2  1 + ↵.

(2)

Next, we claim that both W1 and W2 can be decoded from ˜ n , and thus Y8n and X 3 To see this, we first notice that the removal of v8 and v3 ˜ n , W1 disconnects d1 from {s1 , s2 }. Thus, from Y8n and X 3 can be decoded. Then, W1 can be used to obtain X7n , and from Z structure 1 (Figure 7) and Lemma 3, we know that we ˜ n , which allows us to decode W2 as well. can reconstruct X 6

v7

s1

v8

v10

1

s2

v9

2 v11

d1 v3

v1 3

v6 Fig. 7.

v2

v5

v4

d2

v

Z structures in a network from case C1.

The other two Z structures in Figure 7 (i.e., Z structures 2 and 3) can be used to obtain the inequality D2 + (↵ +

1)  1,

(3)

in the following way. Since the removal of v4 disconnects d2 from {s1 , s2 }, from Y4n we can decode W2 , and thus obtain n X5n and X10 too. Then, Z structure 3 tells us that from Y4n ˜ n . The ↵ fraction of useful information we can reconstruct X 3 transmitted by v3 must come through v11 . Thus, intuitively, n from Y1n , we can reconstruct an ↵ fraction of Y11 . Now, Z n structure 2 tells us that from Y4 we can reconstruct an ↵ ˜ n . Since a fraction of X n (or X ˜ n ) contains fraction of X 9 9 9 n information on W1 , we conclude that, from Y4 , it is possible to decode at least ↵ + 1 degrees-of-freedom of W1 . This implies (3). Finally, by adding inequalities (1), (2) and (3), we obtain 2(D1 + D2 )  3, and therefore D⌃  3/2. The proof for case C2 follows similar steps. An example of such networks is shown in Figure 8. Once again it is possible to achieve 3/2 degrees-of-freedom by allowing the network to operate in two modes and, by finding inequalities that capture the bottlenecks in the network, we can show that D⌃  23 .

s1

d1

s2

d2

Fig. 8.

An example of a network in case C2.

IV. ACKNOWLEDGEMENTS The research of A. S. Avestimehr and I. Shomorony was supported in part by the NSF CAREER award 0953117 and the TRUST NSF Science and Technology Center. R EFERENCES [1] L. R. Ford and D. R. Fulkerson, “Maximal flow through a network,” Canadian Journal of Mathematics, 1956. [2] R. Ahlswede, N. Cai, S.-Y. R. Li, and R. W. Yeung, “Network information flow,” IEEE Trans. on Information Theory, 2000. [3] S.-Y. Li, R. W. Yeung, and N. Cai, “Linear network coding,” IEEE Trans. on Info. Theory, Feb. 2003. [4] R. Koetter and M. M´edard, “An algebraic approach to network coding,” IEEE/ACM Trans. Netw., vol. 11, no. 5, pp. 782–795, 2003. [5] S. Avestimehr, S. Diggavi, and D. Tse, “Wireless network information flow: a deterministic approach,” IEEE Transactions on Information Theory, vol. 57, no. 4, April 2011. [6] A. S. Motahari and A. K. Khandani, “Capacity bounds for the Gaussian interference channel,” IEEE Trans. on Info. Theory, Feb. 2009. [7] X. Shang, G. Kramer, and B. Chen, “A new outer bound and the noisyinterference sum–rate capacity for Gaussian interference channels,” IEEE Trans. on Info. Theory, 2009. [8] V. S. Annapureddy and V. V. Veeravalli, “Gaussian interference networks: Sum capacity in the low-interference regime and new outer bounds on the capacity region,” IEEE Transactions on Information Theory, vol. 55, no. 7, pp. 3032–3050, July 2009. [9] A. E. Gamal and M. Costa, “The capacity region of a class of deterministic interference channels,” IEEE Transactions on information Theory, vol. 28, no. 2, pp. 343–346, March 1982. [10] R. Etkin, D. Tse, and H. Wang, “Gaussian interference channel capacity to within one bit,” IEEE Transactions on Information Theory, vol. 54, no. 12, pp. 5534–5562, December 2008. [11] G. Bresler, A. Parekh, and D. Tse, “The approximate capacity of the many-to-one and one-to-many Gaussian interference channels,” Proc. of Allerton Conference, 2007. [12] V. Cadambe and S. Jafar, “Interference alignment and spatial degrees of freedom for the k user interference channel,” IEEE International Conference on Communications, 2008. [13] A. S. Motahari, S. O. Gharan, and A. K. Khandani, “Real interference alignment with real numbers,” Submitted to IEEE Trans. on Info. Theory. [14] M. A. Maddah-Ali, S. A. Motahari, and A. K. Khandani, “Communication over mimo x channels: Interference alignment, decomposition, and performance analysis,” IEEE Transactions on Information Theory, 2008. [15] T. C. Hu, “Multi-commodity network flows,” Operations Research, vol. 11, no. 3, pp. 344–360, 1963. [16] C. C. Wang and N. B. Shroff, “Beyond the butterfly: A graph-theoretic characterization of the feasibility of network coding with two simple unicast sessions.” ISIT, 2007. [17] S. Shenvi and B. K. Dey, “A simple necessary and sufficient condition for the double unicast problem,” in Proceedings of ICC, 2010. [18] K. Cai, K. B. Letaief, P. Fam, and R. Feng, “On the solvability of 2-pair unicast networks — a cut-based characterization,” arXiv, 2010. [19] S. Mohajer, S. N. Diggavi, C. Fragouli, and D. Tse, “Approximate capacity of gaussian interference-relay networks with weak cross links,” arXiv, 2010. [20] T. Gou, S. Jafar, S.-W. Jeon, and S.-Y. Chung, “Aligned interference neutralization and the degrees of freedom of the 2 ⇥ 2 ⇥ 2 interference channel,” arXiv, 2010. [21] I. Shomorony and A. S. Avestimehr, “Two-unicast wireless networks: Characterizing the sum degrees of freedom,” Submitted to IEEE Trans. on Info. Theory. Available on ArXiv:1102.2498v2, 2011.