#Undirected Flow With  Lower Bounds


    # Create a minimization problem

    # Decision variables

    # Objective function

    # Constraints

    # Lower bound constraints

    # Solve the problem

    # Retrieve the solution

    # Return the objective value and flow dictionary

# Example usage:

# Add edges with lower bounds

# Set the lower bounds for edges



if __name__ == '__main__':
    from pulp import *
    def solve_undirected_flow_with_lower_bounds(graph, lower_bounds):
        prob = LpProblem("Undirected Flow", LpMinimize)
        flow_vars = LpVariable.dicts("Flow", graph.edges(), lowBound=0)
        prob += lpSum(flow_vars[edge] for edge in graph.edges())
        for node in graph.nodes():
            inflow = lpSum(flow_vars[(i, node)] for i in graph.neighbors(node))
            outflow = lpSum(flow_vars[(node, j)] for j in graph.neighbors(node))
            prob += inflow - outflow == 0
        for edge in graph.edges():
            prob += flow_vars[edge] >= lower_bounds[edge]
        prob.solve()
        flow_dict = {(edge[0], edge[1]): flow_vars[edge].varValue for edge in graph.edges()}
        return value(prob.objective), flow_dict
    graph = nx.Graph()
    graph.add_edge('A', 'B', lower_bound=2)
    graph.add_edge('A', 'C', lower_bound=3)
    graph.add_edge('B', 'C', lower_bound=1)
    graph.add_edge('B', 'D', lower_bound=0)
    graph.add_edge('C', 'D', lower_bound=2)
    lower_bounds = {('A', 'B'): 2, ('A', 'C'): 3, ('B', 'C'): 1, ('B', 'D'): 0, ('C', 'D'): 2}
    objective_value, flow_dict = solve_undirected_flow_with_lower_bounds(graph, lower_bounds)
    print("Objective Value:", objective_value)
    print("Flow on Edges:")
    for edge, flow in flow_dict.items():
        print(f"{edge[0]} - {edge[1]}: {flow}")
