#Path Constrained Network Flow


    # Create a new graph with auxiliary nodes for the path constraints

    # Add auxiliary nodes and edges to represent the path constraints

    # Add auxiliary edges from source to the first nodes of the path constraints

    # Add auxiliary edges from the last nodes of the path constraints to the sink

    # Find the maximum flow in the augmented graph

    # Check if the flow satisfies the path constraints


# Example usage:

# Add edges with capacities





if __name__ == '__main__':
    import networkx as nx
    from itertools import combinations
    def has_path_constrained_flow(graph, source, sink, path_constraints):
        augmented_graph = graph.copy()
        for constraint in path_constraints:
            auxiliary_node = 'AUX_' + '_'.join(constraint)
            augmented_graph.add_node(auxiliary_node)
            for u, v in combinations(constraint, 2):
                augmented_graph.add_edge(u, auxiliary_node, capacity=float('inf'))
                augmented_graph.add_edge(auxiliary_node, v, capacity=float('inf'))
        for constraint in path_constraints:
            augmented_graph.add_edge(source, constraint[0], capacity=float('inf'))
        for constraint in path_constraints:
            augmented_graph.add_edge(constraint[-1], sink, capacity=float('inf'))
        flow_value, flow_dict = nx.maximum_flow(augmented_graph, source, sink)
        for constraint in path_constraints:
            auxiliary_node = 'AUX_' + '_'.join(constraint)
            if flow_dict[source][constraint[0]] != 1 or flow_dict[constraint[-1]][sink] != 1 or flow_dict[auxiliary_node][constraint[0]] != 1:
                return False
        return True
    graph = nx.DiGraph()
    graph.add_edge('A', 'B', capacity=5)
    graph.add_edge('A', 'C', capacity=3)
    graph.add_edge('B', 'C', capacity=2)
    graph.add_edge('B', 'D', capacity=4)
    graph.add_edge('C', 'D', capacity=3)
    source = 'A'
    sink = 'D'
    path_constraints = [['A', 'B', 'D'], ['A', 'C', 'D']]
    has_path_flow = has_path_constrained_flow(graph, source, sink, path_constraints)
    print("Has feasible flow satisfying the path constraints:", has_path_flow)
