On the Feasibility of Precoding-Based Network Alignment for Three Unicast Sessions

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1 On the Feasibility of Precoding-Based Network Alignment for Three Unicast Sessions Chun Meng, Abinesh Ramakrishnan, Athina Markopoulou, Syed Ali Jafar Department of Electrical Engineering and Computer Science University of California, Irvine {cmeng1, abinesh.r, athina, Abstract We consider the problem of network coding across three unicast sessions over a directed acyclic graph, when each session has min-cut one. Previous work by Das et al. adapted a precoding-based interference alignment technique, originally developed for the wireless interference channel, specifically to this problem. We refer to this approach as precoding-based network alignment (PBNA). Similar to the wireless setting, PBNA asymptotically achieves half the minimum cut; different from the wireless setting, its feasibility depends on the graph structure. Das et al. provided a set of feasibility conditions for PBNA with respect to a particular precoding matrix. However, the set consisted of an infinite number of conditions, which is impossible to check in practice. Furthermore, the conditions were purely algebraic, without interpretation with regards to the graph structure. In this paper, we first prove that the set of conditions provided by Das. et al are also necessary for the feasibility of PBNA with respect to any precoding matrix. Then, using two graph-related properties and a degree-counting technique, we reduce the set to just four conditions. This reduction enables an efficient algorithm for checking the feasibility of PBNA on a given graph. I. INTRODUCTION Network coding was originally introduced to maximize the rate of a single multicast session over a network [1] [2] [3]. However, network coding across different sessions, which includes multiple unicasts as a special case, is a well-known open problem. For example, finding linear network codes for multiple unicasts is NP-hard [4]. Thus, suboptimal, heuristic approaches, such as linear programming [5] and evolutionary approaches [6], are typically used. Moreover, while it has been shown that scalar or vector linear network codes might be insufficient to achieve the optimal rate [7], only approximation methods [8] exist to characterize the rate region for this setting. In this paper, we consider the simplest inter-session linear network coding scenario: three unicast sessions over a directed acyclic graph, each session with min-cut one. Das et al. [9] applied a precoding-based interference alignment technique, originally developed by Cadambe and Jafar [10] for wireless interference channel, to this problem; we refer to this technique as precoding-based network alignment (PBNA). In a nutshell, PBNA (i) simulates a wireless channel through random network coding [3] in the middle of the network and (ii) applies interference alignment at the edge, i.e., via precoding at the sources and decoding at the receivers. This way, it greatly simplifies the network code design, while it guarantees that each unicast session asymptotically achieves a rate equal to half of its minimum cut [9]. An important difference from the wireless interference channel is that, in our problem, there may be dependencies between elements of the transfer matrix introduced by the graph structure, which make PBNA infeasible in some networks [11]. As a first step, [9] provided a set of feasibility conditions for PBNA, and proved they are sufficient for the feasibility of PBNA with respect to a particular precoding matrix. One important limitation is that the set consists of an infinite number of conditions, which makes it impossible to check in practice. Another limitation is the lack of consideration of graph structures, which is the reason for the significant redundancy in the set. Ramakrishnan et al. [11] conjectured that the infinite set of conditions can be reduced to just two conditions. Han et al. [12] proved that the conjecture holds for three symbol extensions; however, this result cannot be generalized beyond three symbol extensions. In this paper, we make the following contributions. First, we prove that the set of conditions provided in [9] are also necessary for the feasibility of PBNA with respect to any valid precoding matrix. Then, by using a simple degree-counting technique and two graph-related properties, we greatly reduce the set to just four conditions; two of them turn out to have an intuitive interpretation in terms of graph structure. Finally, we present an efficient algorithm for checking the four conditions. The rest of this paper is organized as follows. In Section II, we present the problem formulation. In Section III, we summarize our main results. In Section IV, we discuss the graph-related properties that are key to the simplification of the conditions. In Section V, we prove and discuss our main results regarding the feasibility condition of PBNA. In Section VI, we present an algorithm for checking the condition. In Section VII, we conclude the paper. II. PROBLEM FORMULATION The network is a delay-free directed acyclic graph, denoted by G = (V, E), where V is the set of nodes and E the set of edges. Without loss of generality, each edge has capacity one, i.e., can transmit one symbol of finite field F 2 m in a unit time. For the ith unicast session (i {1, 2, 3}), let s i and d i be its sender and receiver respectively, and R i its transmission rate. Every edge e E represents an error free channel. We assume that the minimum cut between s i and d i is one. Let X i be the source symbol transmitted at s i and Z i be the symbol received at d i. We further extend G as follows: For the ith unicast session (i {1, 2, 3}), we add a virtual sender s i and a virtual receiver d i and two edges σ i = (s i, s i) and τ i = (d i, d i ). The extended graph is denoted by G = (V, E ). For e E, let head(e) and tail(e) denote its head and tail respectively.

2 In the middle of the network, we employ random network coding [3]. The symbol transmitted along e E is a linear combination of incoming symbols at tail(e). For e, e E such that head(e ) = tail(e), let x e e be a variable, which takes values from F 2 m and represents the coding coefficient used to combine the incoming symbol along e into the symbol along e. We group all coding coefficients x e e s into a vector x, called the coding vector of G. The network acts as a linear system: the output at d i is a mixture of source symbols, Z i = 3 j=1 m ij(x)x j, where m ij (x) F 2 m[x] is the transfer function from s j to d i and can be written as follows [2]: m ij (x) = P P ij t(p ), where P ij is the set of paths from s j to d i, and t(p ) is the product of coding coefficients along path P. We assume that all m ij (x) s are non-zeros, which is the most challenging case. Indeed, as shown in [13], when some m ij (x) (i j) is zero, the feasibility condition of PBNA is significantly simplified due to reduced number of interferences. At the edge of the network, we apply interference alignment [9] [10] via precoding at senders and decoding at receivers. Let X i = (Xi 1,, Xki i ) T denote the input vector at sender s i, where k i = n + 1 for i = 1 and k i = n otherwise. We use precoding matrix V i to encode X i into 2n + 1 symbols, which are then transmitted via 2n+1 uses of the network (time slots). The output vector at d i is Z i = (Zi 1,, Z2n+1 i ) T = 3 j=1 M ijv j X j, where M ij is a (2n+1) (2n+1) diagonal matrix with the (k, k) element being m ij (x k ), where x k represents the coding vector for the kth use of the network. V 1 is a (2n+1) (n+1) matrix, and V 2, V 3 are both (2n+1) n matrices. V i can still contain indeterminate variables. Let ξ denote the vector of all variables in x 1,, x 2n+1 and V 1, V 2, V 3. PBNA requires that the following conditions are satisfied for some values of ξ [10]: A 1 : span(m 12 V 2 ) = span(m 13 V 3 ) A 2 : span(m 23 V 3 ) span(m 21 V 1 ) A 3 : span(m 32 V 2 ) span(m 31 V 1 ) B 1 : rank(m 11 V 1 M 12 V 2 ) = 2n + 1 B 2 : rank(m 21 V 1 M 22 V 2 ) = 2n + 1 B 3 : rank(m 31 V 1 M 33 V 3 ) = 2n + 1 Condition A i guarantees that all the interferences at d i are aligned, i.e., mapped into the same linear space; while condition B i ensures that all source symbols for the ith unicast session can be decoded. These conditions ensure that we can achieve a rate tuple (R 1, R 2, R 3 ) = R n = ( n+1 2n+1, n 2n+1, n 2n+1 ), which approaches ( 1 2, 1 2, 1 2 ) as n. In this case, we say that R n is NA-feasible. Previous work [9] [11] [12] only considered the feasibility of PBNA under a particular precoding matrix, i.e., V1 in Eq. (3), which was first introduced in [10]. To address this limitation and characterize the feasibility of PBNA for any precoding matrix, we reformulate A 1, A 2, A 3 and B 1, B 2, B 3 without any assumption about the structure of precoding matrix. First, we reformulate A 1, A 2, A 3 as: 1 : M 12 V 2 = M 13 V 3 A 2 : M 23 V 3 = M 21 V 1 B 3 : M 32 V 2 = M 31 V 1 C where A is an n n invertible matrix, and B and C are both (n + 1) n matrices with rank n. A 1, A 2, A 3 can be further condensed into a single condition: where T = M 12 M 1 21 M 23M 1 B 1, B 2, B 3 are reformulated as: TV 1 C = V 1 BA (1) 32 M 31M 1 13 B 1 : ψ 1 (ξ) = det(v 1 P 1 V 1 C) 0 B 2 : ψ 2 (ξ) = det(v 1 P 2 V 1 C) 0. Finally, conditions B 3 : ψ 3 (ξ) = det(v 1 P 3 V 1 CA 1 ) 0 where P 1 = M 31 M 1 11 M 12M 1 32, P 2 = M 31 M 1 21 M 22M 1 32, and P 3 = M 12 M 1 32 M 33M 1 13, and ψ 1(ξ), ψ 2 (ξ), ψ 3 (ξ) are rational functions in the field F 2 m(ξ). Define ψ(ξ) = 3 i=1 ψ i(ξ). We assume that F 2 m is sufficiently large such that if ψ(ξ) is a non-zero rational function, there are values to ξ, denoted by ξ 0, such that ψ(ξ 0 ) 0. We also define the following rational functions: p 1(x) = m31(x)m12(x) m 11(x)m 32(x) p 3(x) = m12(x)m33(x) m 32(x)m 13(x) p2(x) = m31(x)m22(x) m 21(x)m 32(x) = m31(x)m12(x)m23(x) m 21(x)m 32(x)m 13(x) Clearly, p i (x) and form the elements along the diagonals of P i and T respectively. Hence, the following lemma holds: Lemma 1: R n is NA-feasible if and only if 1) Eq. (1) is satisfied, and 2) B 1, B 2, B 3 are satisfied. The fundamental design problem in PBNA is to find V 1 such that all the conditions in Lemma 1 are satisfied. Indeed, the major restriction on V 1 comes from Eq. (1). As shown in [9], the construction of V 1 depends on whether is constant. When is constant, and thus T = I 2n+1, we set C = BA. Therefore, any arbitrary V 1 can satisfy Eq. (1). In contrast, when is not constant, we can no longer choose V 1 freely. [10] proposed the following precoding matrix, which has also been used by most of recent work [9] [11] [12]: (2) V 1 = (w Tw T n w) (3) where w is a column vector of 2n + 1 ones. Meanwhile, we set A = I n, C consists of the left n columns of I n+1, and B the right n columns of I n+1 ; this construction satisfies Eq. (1). Note that the form of V 1 is determined by A, B and C. With different A, B and C, we can derive different V 1 ; therefore the choice of V 1 is not limited to just V1. As observed in [11], graphs can introduce dependence between transfer functions 1 so that PBNA may be infeasible. This is a fundamental difference compared to wireless interference channel, where channel gains can change independently and interference alignment is always feasible. Fig. 1 Dependence here means that one transfer function (namely m ii (x), corresponding to signal for the ith unicast flow) can be written as a rational function of other transfer (interference) functions. The exact functional form is dictated by Eq. (5) or Eq. (6).

3 (a) p 1 (x) = p 2 (x) = p 3 (x) = = 1 (b) p 1 (x) = +1 Fig. 1: Examples of graphs where PBNA is infeasible 1 depicts some examples of graphs where PBNA is infeasible. In Fig. 1(a), p i (x) = = 1 for i {1, 2, 3}, thus P i = I 2n+1, implying B 1, B 2, B 3 are all violated. In Fig. 1(b), p 1 (x) = +1, which also violates B 1. This example shows that the conjecture proposed by Ramakrishnan et al. [11] doesn t hold beyond three symbol extensions. Moreover, by exchanging s 1 s 2 and d 1 d 2, we obtain another counter example, where p 2 (x) = 1 +, violating B 2. As a first step, [9] proposed the following set of conditions for PBNA. 2 For i [1 : 3], { } f() p i (x) / g() : f(z), g(z) F 2m[z], g(z) 0 (4) In [9], it was proved that if Eq. (4) is satisfied, we can use V 1 to asymptotically achieve half rate through infinite number of uses of the network. Unfortunately, since Eq. (4) contains an infinite number of conditions, it is impractical to verify. Moreover, since only one particular matrix was considered in [9], Eq. (4) was only shown to be sufficient for PBNA. III. OVERVIEW OF MAIN RESULTS We now state our main results; proofs are deferred to Section V and [13]. Since the construction of V 1 depends on whether is constant, we distinguish two cases. A. Is Constant In this case, we can choose V 1 freely, and thus the feasibility condition of PBNA can be significantly simplified, as stated in the following theorem: Theorem 1: Assume is constant. R n is NA-feasible if and only if p i (x) is not constant for each i {1, 2, 3}. B. Is Not Constant In this case, we cannot choose V 1 freely. Using similar technique as in [9], we can rewrite Eq. (4) as follows: 3 { f() p i (x) / S n = g() : f(z), g(z) F 2 m[z], f(z)g(z) 0, gcd(f(z), g(z)) = 1, } d f n, d g n 1 i {1, 2, 3} Note that, in contrast to Eq. (4), the above set of conditions guarantee that R n is NA-feasible for a fixed value of n. 2 There is actually a small difference between Eq. (4) and the original formulation in [9], in which p 1 (x) is replaced by 1/p 1 (x). It is easy to see that the two are equivalent. 3 Notation: For two polynomials f(x) and g(x), let gcd(f(x), g(x)) denote their greatest common divisor, and d f the degree of f(x). (5) Fig. 2: The construction of H (in the proof of Linearization Property) is enabled by Lemma 2 (P 1 is disjoint with P 2). Next, we show that Eq. (5) is also necessary for the feasibility of PBNA with respect to any V 1 satisfying the conditions of Lemma 1. Theorem 2: Assume is not constant. R n is NAfeasible if and only if for each i {1, 2, 3}, p i (x) / S n. Finally, we greatly reduce S n to just four rational functions: Theorem 3 (The Main Theorem): Assume is not constant. R n is NA-feasible if and only if for each i {1, 2, 3}, the following condition is satisfied: p i (x) / S = { 1,, + 1, } + 1 The basic idea behind the Main Theorem is that we can compare the degree of a variable in p i (x) with that of a rational function in S n. This technique enables us to reduce S n to the form { a0+a1 b 0+b 1 }. Thus, we only need to consider a finite number of rational functions, namely S in Eq. (6). This enables an efficient algorithm for checking the feasibility of PBNA. The key for enabling this reduction lies in two graphrelated properties, which we refer to as Linearization Property and Square-Term Property, as described in the next section. IV. GRAPH-RELATED PROPERTIES Our key intuition is that p i (x) is not an arbitrary function but depends on transfer functions, as specified in Eq. (2). Therefore, p i (x) have special algebraic properties, which can be exploited to simplify Eq. (5). First note that all p i (x) s are of the following general form: h(x) = m ab(x)m pq (x), a, b, p, q {1, 2, 3}, a p, b q. m aq (x)m pb (x) Furthermore, each path pair in P ab P pq contributes a term in m ab (x)m pq (x), and each path pair in P aq P pb contributes a term in m aq (x)m pb (x): m ab (x)m pq (x) = (P 1,P 2) P ab P pq t(p 1 )t(p 2 ) m aq (x)m pb (x) = (P 3,P 4) P aq P pb t(p 3 )t(p 4 ) A. Linearization Property First, consider the following lemma, which provides an easy way to check whether p i (x) / {1, } (as in Section VI). The intuition is that we can multicast two symbols from s b, s q to d a, d p by network coding if and only if the minimum cut separating s b, s q from d a, d p is greater than one [2]. Lemma 2: m ab (x)m pq (x) m aq (x)m pb (x) if and only if there is disjoint path pair (P 1, P 2 ) P ab P pq or (P 3, P 4 ) P aq P pb. (6)

4 Fig. 3: Illustration of Square-Term Property. A term with x 2 ee introduced by (P 1, P 2) in the numerator of h(x) equals another term introduced by (P 3, P 4) in the denominator of h(x). The first graph-related property states that p i (x) can be transformed into its simplest non-trivial form (i.e., a linear function or the inverse of a linear function). More formally: Lemma 3 (Linearization Property): Let h(x) = m ab (x)m pq(x) m aq(x)m pb (x) = u(x) v(x) such that gcd(u(x), v(x)) = 1. Assume h(x) is not constant. Then, for sufficiently large m, we can assign values to x other than a variable x ee such that u(x) and v(x) are transformed into either u(x ee ) = c 1 x ee + c 0, v(x ee ) = c 2 or u(x ee ) = c 2, v(x ee ) = c 1 x ee + c 0, where c 0, c 1, c 2 are constants in F 2 m, and c 1 c 2 0. Proof: Due to lack of space, we provide the proof outline and refer the reader to [13] for the complete proof. The key is to find a subgraph H and consider h(x) restricted to H, i.e., h(x H ) = m ab(x H )m pq(x H ) m aq(x H )m pb (x H ), where x H represents the coding vector of H. In addition, we require that some variable x ee appears exclusively in the numerator or the denominator of h(x H ). Thus, by assigning values to x H other than x ee, we can transform h(x H ) into a linear function or the inverse of a linear function in terms of x ee. Since h(x H ) can be acquired through a partial assignment to x, this transformation also holds for the complete graph G. We then consider how to find such H. By Lemma 2, there exists a disjoint path pair, say (P 1, P 2 ) P ab P pq. Pick another path pair (P 3, P 4 ) P aq P pb and let H be the subgraph induced by P 1 P 2 P 3 P 4. This procedure is illustrated in Fig. 2. As shown in this figure, the structure of H can be one of three cases, depending on the relative positions of the branching points and joining points between P 1 P 4 (see [13]). A key observation is that for each case, there must be e, e P 1 P 2 such that x ee appears exclusively in the numerator of h(x H ). Similarly, if there exists disjoint path pair (P 3, P 4 ) P aq P pb, there exists x ee which appears exclusively in the denominator of h(x H ). B. Square-Term Property The second graph-related property is stated in Lemma 4: the coefficient of x 2 ee in the numerator of h(x) equals its counterpart in the denominator of h(x). Thus, if x 2 ee appears in the numerator of h(x) under some assignment to x, it must also appear in the denominator of h(x), and vice versa. Lemma 4 (Square-Term Property): For a coding variable x ee, let f 1 (x) and f 2 (x) be the coefficients of x 2 ee in m ab (x)m pq (x) and m aq (x)m pb (x) respectively. Then f 1 (x) = f 2 (x). Proof: For any x ee, define Q 1 = {(P 1, P 2 ) P ab P pq : x 2 ee t(p 1)t(P 2 )} and Q 2 = {(P 3, P 4 ) P aq P pb : x 2 ee t(p 3 )t(p 4 )}. Consider a path pair (P 1, P 2 ) Q 1. Since the degree of x ee in t(p 1 ) and t(p 2 ) is at most one, we must have x ee t(p 1 ) and x ee t(p 2 ). Thus e, e P 1 P 2. Let P1 1, P1 2 be the parts of P 1 before e and after e respectively. Similarly, define P2 1 and P2 2. Then construct two new paths: P 3 = P2 1 {e, e } P1 2 and P 4 = P1 1 {e, e } P2 2 (see Fig. 3). Clearly, t(p 1 )t(p 2 ) = t(p 3 )t(p 4 ), and thus (P 3, P 4 ) Q 2. The above method establishes a one-to-one mapping φ : Q 1 Q 2, such that for φ((p 1, P 2 )) = (P 3, P 4 ), t(p 1 )t(p 2 ) = t(p 3 )t(p 4 ). Hence, f 1 (x) = 1 x (P 2 ee 1,P 2) Q 1 t(p 1 )t(p 2 ) = 1 x (P 2 ee 3,P 4) Q 2 t(p 3 )t(p 4 ) = f 2 (x). V. FEASIBILITY CONDITION OF PBNA In this section, we provide the proofs of Theorems 1, 2 and the proof outline of the Main Theorem. Due to lack of space, details are deferred to [13]. A. Is Constant Proof of Theorem 1: As shown in [9], when is constant and thus T = I 2n+1, we set C = BA and V 1 = (θ ij ) (2n+1) (n+1), where θ ij is an arbitrary variable, such that Eq. (1) is satisfied. Hence, if p i (x) is not constant, ψ i (ξ) 0 and B i is satisfied. Thus R n is NA-feasible. Conversely, if p i (x) is constant and P i = I 2n+1, B i is violated, and R n is not NA-feasible. B. Is Not Constant The general form of V 1 that satisfies Eq. (1) is as follows: Lemma 5: Any V 1 satisfying Eq. (1) has the form V 1 = GV1F, where V1 is defined in Eq. (3), F is an (n+1) (n+1) matrix, and G is a (2n + 1) (2n + 1) diagonal matrix, with the (i, i) element being f i (η(x i )), where f i (z) is a non-zero rational function in F 2 m(z). Moreover, the (n + 1)th row of FC and the 1st row of FBA are both zero vectors. Lemma 5 indicates that there is a direct relation between V1 and the general form of V 1, which we use to prove that Eq. (5) is also necessary for the feasibility of PBNA. Proof of Theorem 2: The sufficiency of Eq. (5) was proved in [9]. Now assume p i (x) = f() g() S n, where f(z) = n k=0 a kz k and g(z) = n 1 k=0 b kz k. We will prove that for any V 1 satisfying Eq. (1), B i cannot be satisfied, thus R n is not NA-feasible. Apparently, if rank(v 1 ) < n + 1, B i is violated. Thus, in the rest of this proof, we assume rank(v 1 ) = n + 1. By Lemma 5, V 1 = GV1F, where F is an (n + 1) (n + 1) invertible matrix. The jth row of V 1 is r j = f j (η(x j ))(1, η(x j ),, η n (x j ))F. Since the (n + 1)th row of FC is zero, we have r j C = f j (η(x j ))(1, η(x j ),, η n 1 (x j ))H, where H consists of the top n rows of FC and rank(h) = n. Let a = (a 0, a 1,, a n ) T and b = (b 0, b 1,, b n 1 ) T. For i = 1, 2, 3, we define a = F 1 a and b = H 1 b. It follows r j a = f j (η(x j ))(1, η(x j ),, η n (x j ))a = f j (η(x j ))f(η(x j )) = f j (η(x j ))p i (x j )g(η(x j )) = p i (x j )f j (η(x j ))(1, η(x j ),, η n 1 (x j ))b = p i (x j )f j (η(x j ))(1, η(x j ),, η n 1 (x j ))Hb = p i (x j )r j Cb

5 Hence, the columns of (V 1 P i V 1 C) are linearly dependent, violating B i. Likewise, we can prove the case i = 3. For the proof of the Main Theorem, we need to rearrange the ratio of rational functions f() g() in Eq. (6) to a ratio of coprime polynomials with variables x. To this end, we use a property of polynomials stated in the following lemma. Lemma 6: Let F be a field. z is a variable and y = (y 1, y 2,, y k ) is a vector of variables. Consider four nonzero polynomials f(z), g(z) F[z] and s(y), t(y) F[y], such that gcd(f(z), g(z)) = 1 and gcd(s(y), t(y)) = 1. Denote d = max{d f, d g }. Define two polynomials in F[y]: α(y) = f( s(y) t(y) )td (y) and β(y) = g( s(y) t(y) )td (y). Then gcd(α(y), β(y)) = 1. We now have all the ingredients to outline the proof of the Main Theorem. The complete proof is described in [13]. Proof Outline of the Main Theorem: The necessity of Eq. (6) follows directly from Theorem 2. We only need to prove the sufficiency of Eq. (6). Our objective is to prove that if p i (x) / S, p i (x) / S n, and thus by Theorem 2, R n is NA-feasible. By contradiction, assume p i (x) = f() g(), where f(z), g(z) F 2 m[z] and gcd(f(z), g(z)) = 1. Let p i (x) = u(x) s(x) v(x) and = t(x) such that gcd(u(x), v(x)) = gcd(s(x), t(x)) = 1. Denote d = max{d f, d g }. Define α(x) = f()t d (x) and β(x) = g()t d (x). The proof consists of the following two steps: Step I: We apply a simple degree-counting technique along with Lemma 3 and 6 to prove that d = 1, and thus f() g() can only be of the form a0+a1 b 0+b 1. By Lemma 6, we have u(x) = α(x) and v(x) = β(x). According to Lemma 3, we assign values to x, such that either u(x ee ) = c 1 x ee + c 0 = α(x ee ), v(x ee ) = c 2 = β(x ee ) or u(x ee ) = c 2 = α(x ee ), v(x ee ) = c 1 x ee + c 0 = β(x ee ). Note in either case, max{d u, d v } = 1. Hence, by comparing the degrees of these equations, we can prove that d = 1. Step II: We use Lemma 3 and 4 to rule out other cases. For example, consider the case p 1 (x) = a0+a1 b 1, where a 0 a 1 b 1 0. Define q 1 (x) = p 1(x) = m11(x)m23(x) m. 13(x)m 21(x) Under the above assignment, we can derive q 1 (x ee ) = a 0c 2 2 b 1c 2 1 x2 ee a 1c 1c 2x ee +b. However, by Lemma 4, 1c 2 x2 0 a1c0c2 ee must appear in both the denominator and numerator of q 1 (x ee ), which contradicts the formulation of q 1 (x ee ). Hence, p 1 (x) a0+a1 b 1. The other cases follow along similar lines. In summary, we show that if p i (x) / S, p i (x) / S n, and thus R n is NA-feasible. VI. CHECKING THE FEASIBILITY OF PBNA For a given graph, checking the feasibility of PBNA is now reduced to checking whether p i (x) does not have any of the four simple functional forms in Eq. (6), for i {1, 2, 3}. This is a multivariate polynomial identity testing problem. To check whether p i (x) 1, we use Ford-Fulkerson Algorithm, as per Lemma 2. To check whether p i (x), we define q i (x) = p and consider q i(x) i(x) 1. Therefore, Ford- Fulkerson Algorithm can be used to check this condition as well. For the last two conditions (p i (x) / {1+, +1 }), it is still not clear what is their interpretation in terms of graph structure. A counter example is shown in Fig. 1(b). Nevertheless, we can still check the conditions by evaluating the rational functions through T random tests: for k = 1 to T do Assign random values to x, denoted by x 0 η(x If p i(x 0) / {1 + η(x 0), 0 ) η(x 0 }, return success )+1 end for Return failure (i.e., B i is violated) Using Lemma 4 of [3], we can upper-bound the probability of error by 2[1 (1 1 2 ) L ] T, where L is the maximum distance m 2 from any sender to any receiver in the network. Thus, the error can be made arbitrarily small for sufficiently large m and T. The running time of the algorithm is O(T E D in ), where D in is the maximum in-degree of any node in the network. VII. CONCLUSION In this paper, we study the feasibility of PBNA for three unicast sessions. 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