Max-flow Min-cut Theorem - Linear Program Formulation

Linear Program Formulation

The max-flow problem and min-cut problem can be formulated as two primal-dual linear programs.

Max-flow (Primal)

Min-cut (Dual)

maximize

minimize

subject to

\begin{array}{rclr} f_{ij} & \leq & c_{ij} & (i, j) \in E \\ \sum_{j: (j, i) \in E} f_{ji} - \sum_{j: (i, j) \in E} f_{ij} & \leq & 0 & i \in V, i \neq s,t \\ \nabla_s + \sum_{j: (j, s) \in E} f_{js} - \sum_{j: (s, j) \in E} f_{sj} & \leq & 0 & \\ \nabla_t + \sum_{j: (j, t) \in E} f_{jt} - \sum_{j: (t, j) \in E} f_{tj} & \leq & 0 & \\ f_{ij} & \geq & 0 & (i, j) \in E\\ \end{array}

subject to

\begin{array}{rclr} d_{ij} - p_i + p_j & \geq & 0 & (i, j) \in E \\ p_s & = & 1 & \\ p_t & = & 0 & \\ p_i & \geq & 0 & i \in V \\ d_{ij} & \geq & 0 & (i, j) \in E \end{array}

The equality in the max-flow min-cut theorem follows from the strong duality theorem in linear programming, which states that if the primal program has an optimal solution, x*, then the dual program also has an optimal solution, y*, such that the optimal values formed by the two solutions are equal.

Read more about this topic:  Max-flow Min-cut Theorem

Famous quotes containing the words program and/or formulation:

    “Here also was made the novelty ‘Chestnut Bell’ which enjoyed unusual popularity during the gay nineties when every dandy jauntily wore one of the tiny bells on the lapel of his coat, and rang it whenever a story-teller offered a ‘chestnut.’”
    —Administration for the State of Con, U.S. public relief program (1935-1943)

    “Art is an experience, not the formulation of a problem.”
    —Lindsay Anderson (b. 1923)