Skip to main content
`Ctrl`+`K`

# Source code for networkx.algorithms.flow.mincost

    """
    Minimum cost flow algorithms on directed connected graphs.
    """

    __all__ = ["min_cost_flow_cost", "min_cost_flow", "cost_of_flow", "max_flow_min_cost"]

    import networkx as nx

    @nx._dispatchable(
        node_attrs="demand", edge_attrs={"capacity": float("inf"), "weight": 0}
    )
    def min_cost_flow_cost(G, demand="demand", capacity="capacity", weight="weight"):
        r"""Find the cost of a minimum cost flow satisfying all demands in digraph G.

        G is a digraph with edge costs and capacities and in which nodes
        have demand, i.e., they want to send or receive some amount of
        flow. A negative demand means that the node wants to send flow, a
        positive demand means that the node want to receive flow. A flow on
        the digraph G satisfies all demand if the net flow into each node
        is equal to the demand of that node.

        Parameters
        ----------
        G : NetworkX graph
            DiGraph on which a minimum cost flow satisfying all demands is
            to be found.

        demand : string
            Nodes of the graph G are expected to have an attribute demand
            that indicates how much flow a node wants to send (negative
            demand) or receive (positive demand). Note that the sum of the
            demands should be 0 otherwise the problem in not feasible. If
            this attribute is not present, a node is considered to have 0
            demand. Default value: 'demand'.

        capacity : string
            Edges of the graph G are expected to have an attribute capacity
            that indicates how much flow the edge can support. If this
            attribute is not present, the edge is considered to have
            infinite capacity. Default value: 'capacity'.

        weight : string
            Edges of the graph G are expected to have an attribute weight
            that indicates the cost incurred by sending one unit of flow on
            that edge. If not present, the weight is considered to be 0.
            Default value: 'weight'.

        Returns
        -------
        flowCost : integer, float
            Cost of a minimum cost flow satisfying all demands.

        Raises
        ------
        NetworkXError
            This exception is raised if the input graph is not directed or
            not connected.

        NetworkXUnfeasible
            This exception is raised in the following situations:

                * The sum of the demands is not zero. Then, there is no
                  flow satisfying all demands.
                * There is no flow satisfying all demand.

        NetworkXUnbounded
            This exception is raised if the digraph G has a cycle of
            negative cost and infinite capacity. Then, the cost of a flow
            satisfying all demands is unbounded below.

        See also
        --------
        cost_of_flow, max_flow_min_cost, min_cost_flow, network_simplex

        Notes
        -----
        This algorithm is not guaranteed to work if edge weights or demands
        are floating point numbers (overflows and roundoff errors can
        cause problems). As a workaround you can use integer numbers by
        multiplying the relevant edge attributes by a convenient
        constant factor (eg 100).

        Examples
        --------
        A simple example of a min cost flow problem.

        >>> G = nx.DiGraph()
        >>> G.add_node("a", demand=-5)
        >>> G.add_node("d", demand=5)
        >>> G.add_edge("a", "b", weight=3, capacity=4)
        >>> G.add_edge("a", "c", weight=6, capacity=10)
        >>> G.add_edge("b", "d", weight=1, capacity=9)
        >>> G.add_edge("c", "d", weight=2, capacity=5)
        >>> flowCost = nx.min_cost_flow_cost(G)
        >>> flowCost
        24
        """
        return nx.network_simplex(G, demand=demand, capacity=capacity, weight=weight)[0]

    @nx._dispatchable(
        node_attrs="demand", edge_attrs={"capacity": float("inf"), "weight": 0}
    )
    def min_cost_flow(G, demand="demand", capacity="capacity", weight="weight"):
        r"""Returns a minimum cost flow satisfying all demands in digraph G.

        G is a digraph with edge costs and capacities and in which nodes
        have demand, i.e., they want to send or receive some amount of
        flow. A negative demand means that the node wants to send flow, a
        positive demand means that the node want to receive flow. A flow on
        the digraph G satisfies all demand if the net flow into each node
        is equal to the demand of that node.

        Parameters
        ----------
        G : NetworkX graph
            DiGraph on which a minimum cost flow satisfying all demands is
            to be found.

        demand : string
            Nodes of the graph G are expected to have an attribute demand
            that indicates how much flow a node wants to send (negative
            demand) or receive (positive demand). Note that the sum of the
            demands should be 0 otherwise the problem in not feasible. If
            this attribute is not present, a node is considered to have 0
            demand. Default value: 'demand'.

        capacity : string
            Edges of the graph G are expected to have an attribute capacity
            that indicates how much flow the edge can support. If this
            attribute is not present, the edge is considered to have
            infinite capacity. Default value: 'capacity'.

        weight : string
            Edges of the graph G are expected to have an attribute weight
            that indicates the cost incurred by sending one unit of flow on
            that edge. If not present, the weight is considered to be 0.
            Default value: 'weight'.

        Returns
        -------
        flowDict : dictionary
            Dictionary of dictionaries keyed by nodes such that
            flowDict[u][v] is the flow edge (u, v).

        Raises
        ------
        NetworkXError
            This exception is raised if the input graph is not directed or
            not connected.

        NetworkXUnfeasible
            This exception is raised in the following situations:

                * The sum of the demands is not zero. Then, there is no
                  flow satisfying all demands.
                * There is no flow satisfying all demand.

        NetworkXUnbounded
            This exception is raised if the digraph G has a cycle of
            negative cost and infinite capacity. Then, the cost of a flow
            satisfying all demands is unbounded below.

        See also
        --------
        cost_of_flow, max_flow_min_cost, min_cost_flow_cost, network_simplex

        Notes
        -----
        This algorithm is not guaranteed to work if edge weights or demands
        are floating point numbers (overflows and roundoff errors can
        cause problems). As a workaround you can use integer numbers by
        multiplying the relevant edge attributes by a convenient
        constant factor (eg 100).

        Examples
        --------
        A simple example of a min cost flow problem.

        >>> G = nx.DiGraph()
        >>> G.add_node("a", demand=-5)
        >>> G.add_node("d", demand=5)
        >>> G.add_edge("a", "b", weight=3, capacity=4)
        >>> G.add_edge("a", "c", weight=6, capacity=10)
        >>> G.add_edge("b", "d", weight=1, capacity=9)
        >>> G.add_edge("c", "d", weight=2, capacity=5)
        >>> flowDict = nx.min_cost_flow(G)
        >>> flowDict
        {'a': {'b': 4, 'c': 1}, 'd': {}, 'b': {'d': 4}, 'c': {'d': 1}}
        """
        return nx.network_simplex(G, demand=demand, capacity=capacity, weight=weight)[1]

    @nx._dispatchable(edge_attrs={"weight": 0})
    def cost_of_flow(G, flowDict, weight="weight"):
        """Compute the cost of the flow given by flowDict on graph G.

        Note that this function does not check for the validity of the
        flow flowDict. This function will fail if the graph G and the
        flow don't have the same edge set.

        Parameters
        ----------
        G : NetworkX graph
            DiGraph on which a minimum cost flow satisfying all demands is
            to be found.

        weight : string
            Edges of the graph G are expected to have an attribute weight
            that indicates the cost incurred by sending one unit of flow on
            that edge. If not present, the weight is considered to be 0.
            Default value: 'weight'.

        flowDict : dictionary
            Dictionary of dictionaries keyed by nodes such that
            flowDict[u][v] is the flow edge (u, v).

        Returns
        -------
        cost : Integer, float
            The total cost of the flow. This is given by the sum over all
            edges of the product of the edge's flow and the edge's weight.

        See also
        --------
        max_flow_min_cost, min_cost_flow, min_cost_flow_cost, network_simplex

        Notes
        -----
        This algorithm is not guaranteed to work if edge weights or demands
        are floating point numbers (overflows and roundoff errors can
        cause problems). As a workaround you can use integer numbers by
        multiplying the relevant edge attributes by a convenient
        constant factor (eg 100).

        Examples
        --------
        >>> G = nx.DiGraph()
        >>> G.add_node("a", demand=-5)
        >>> G.add_node("d", demand=5)
        >>> G.add_edge("a", "b", weight=3, capacity=4)
        >>> G.add_edge("a", "c", weight=6, capacity=10)
        >>> G.add_edge("b", "d", weight=1, capacity=9)
        >>> G.add_edge("c", "d", weight=2, capacity=5)
        >>> flowDict = nx.min_cost_flow(G)
        >>> flowDict
        {'a': {'b': 4, 'c': 1}, 'd': {}, 'b': {'d': 4}, 'c': {'d': 1}}
        >>> nx.cost_of_flow(G, flowDict)
        24
        """
        return sum((flowDict[u][v] * d.get(weight, 0) for u, v, d in G.edges(data=True)))

    @nx._dispatchable(edge_attrs={"capacity": float("inf"), "weight": 0})
    def max_flow_min_cost(G, s, t, capacity="capacity", weight="weight"):
        """Returns a maximum (s, t)-flow of minimum cost.

        G is a digraph with edge costs and capacities. There is a source
        node s and a sink node t. This function finds a maximum flow from
        s to t whose total cost is minimized.

        Parameters
        ----------
        G : NetworkX graph
            DiGraph on which a minimum cost flow satisfying all demands is
            to be found.

        s: node label
            Source of the flow.

        t: node label
            Destination of the flow.

        capacity: string
            Edges of the graph G are expected to have an attribute capacity
            that indicates how much flow the edge can support. If this
            attribute is not present, the edge is considered to have
            infinite capacity. Default value: 'capacity'.

        weight: string
            Edges of the graph G are expected to have an attribute weight
            that indicates the cost incurred by sending one unit of flow on
            that edge. If not present, the weight is considered to be 0.
            Default value: 'weight'.

        Returns
        -------
        flowDict: dictionary
            Dictionary of dictionaries keyed by nodes such that
            flowDict[u][v] is the flow edge (u, v).

        Raises
        ------
        NetworkXError
            This exception is raised if the input graph is not directed or
            not connected.

        NetworkXUnbounded
            This exception is raised if there is an infinite capacity path
            from s to t in G. In this case there is no maximum flow. This
            exception is also raised if the digraph G has a cycle of
            negative cost and infinite capacity. Then, the cost of a flow
            is unbounded below.

        See also
        --------
        cost_of_flow, min_cost_flow, min_cost_flow_cost, network_simplex

        Notes
        -----
        This algorithm is not guaranteed to work if edge weights or demands
        are floating point numbers (overflows and roundoff errors can
        cause problems). As a workaround you can use integer numbers by
        multiplying the relevant edge attributes by a convenient
        constant factor (eg 100).

        Examples
        --------
        >>> G = nx.DiGraph()
        >>> G.add_edges_from(
        ...     [
        ...         (1, 2, {"capacity": 12, "weight": 4}),
        ...         (1, 3, {"capacity": 20, "weight": 6}),
        ...         (2, 3, {"capacity": 6, "weight": -3}),
        ...         (2, 6, {"capacity": 14, "weight": 1}),
        ...         (3, 4, {"weight": 9}),
        ...         (3, 5, {"capacity": 10, "weight": 5}),
        ...         (4, 2, {"capacity": 19, "weight": 13}),
        ...         (4, 5, {"capacity": 4, "weight": 0}),
        ...         (5, 7, {"capacity": 28, "weight": 2}),
        ...         (6, 5, {"capacity": 11, "weight": 1}),
        ...         (6, 7, {"weight": 8}),
        ...         (7, 4, {"capacity": 6, "weight": 6}),
        ...     ]
        ... )
        >>> mincostFlow = nx.max_flow_min_cost(G, 1, 7)
        >>> mincost = nx.cost_of_flow(G, mincostFlow)
        >>> mincost
        373
        >>> from networkx.algorithms.flow import maximum_flow
        >>> maxFlow = maximum_flow(G, 1, 7)[1]
        >>> nx.cost_of_flow(G, maxFlow) >= mincost
        True
        >>> mincostFlowValue = sum((mincostFlow[u][7] for u in G.predecessors(7))) - sum(
        ...     (mincostFlow[7][v] for v in G.successors(7))
        ... )
        >>> mincostFlowValue == nx.maximum_flow_value(G, 1, 7)
        True

        """
        maxFlow = nx.maximum_flow_value(G, s, t, capacity=capacity)
        H = nx.DiGraph(G)
        H.add_node(s, demand=-maxFlow)
        H.add_node(t, demand=maxFlow)
        return min_cost_flow(H, capacity=capacity, weight=weight)