Solving the Single Rate 2-Pair Network Coding Problem with the A-Set Equation

The single rate 2-pair network coding problem is investigated. It is shown that a 2-pair network N = (V, E, {s₁, s₂},{t₁, t₂}) is solvable if and only if (A_1,2 ∪ A_2,1)∩(A_1,1 ∩ A_2,2) = 0, where A_i,j is the union of the minimal cuts of the point-to-point network N_ij = (V, E, S_i, t_j). As a result, a cut-based algorithm can be obtained for solving the single rate 2-pair network coding problem.