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; s_1, s_2; t_1; t_2) is solvable if and only if (A_{1,2}?O A_{2,1}) ?? (A_{1,1}?OA_{2,2}) = ?³, where A_{i,j} is the union of the minimal cuts of the point-to-point network N_{i,j} = (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.