Class CapacityScalingMinimumCostFlow<V,​E>

  • Type Parameters:
    V - graph vertex type
    E - graph edge type
    All Implemented Interfaces:
    FlowAlgorithm<V,​E>, MinimumCostFlowAlgorithm<V,​E>

    public class CapacityScalingMinimumCostFlow<V,​E>
    extends java.lang.Object
    implements MinimumCostFlowAlgorithm<V,​E>
    This class computes a solution to a minimum cost flow problem using the successive shortest path algorithm with capacity scaling. More precisely, this class computes a b-flow of minimum cost, i.e. for each node $v$ in the network the sum of all outgoing flows minus the sum of all incoming flows should be equal to the node supply $b_v$

    The minimum cost flow problem is defined as follows: \[ \begin{align} \mbox{minimize}~& \sum_{e\in \delta^+(s)}c_e\cdot f_e &\\ \mbox{s.t. }&\sum_{e\in \delta^-(i)} f_e - \sum_{e\in \delta^+(i)} f_e = b_e & \forall i\in V\\ &l_e\leq f_e \leq u_e & \forall e\in E \end{align} \] Here $\delta^+(i)$ and $\delta^-(i)$ denote the outgoing and incoming edges of vertex $i$ respectively. The parameters $c_{e}$ define a cost for each unit of flow on the arc $e$, $l_{e}$ define minimum arc flow and $u_{e}$ define maximum arc flow. If $u_{e}$ is equal to CAP_INF, then arbitrary large flow can be sent across the arc $e$. Parameters $b_{e}$ define the nodes demands: positive demand means that a node is a supply node, 0 demand means that it is a transhipment node, negative demand means that it is a demand node. Parameters $b_{e}$, $l_{e}$ and $u_{e}$ can be specified via MinimumCostFlowProblem, graph edge weights are considered to be parameters $c_{e}$, which can be negative.

    This algorithm supports two modes: with and without scaling. An integral scaling factor can be specified during construction time. If the specified scaling factor is less than 2, then the algorithm solves the specified problem using regular successive shortest path. The default scaling factor is DEFAULT_SCALING_FACTOR.

    Essentially, the capacity scaling technique is breaking down the solution of the problem into $O(\log U)$ phases $\left[\Delta_i, \Delta_{i +1}\right],\ \Delta_i = 2^{i}, i = 0, 1, \dots, \log_a(U) - 1$. At each phase the algorithm carries at least $\delta_i$ units of flow. This technique ensures weakly polynomial time bound on the running time complexity of the algorithm. Smaller scaling factors guarantee smaller constant in the asymptotic time bound. The best choice of scaling factor is between $2$ and $16$, which depends on the characteristics of the flow network. Choosing $100$ as a scaling factor is almost equivalent to using the algorithm without scaling. In the case the algorithm is used without scaling, it has pseudo-polynomial time complexity $\mathcal{O}(nU(m + n)\log n)$.

    Currently the algorithm doesn't support undirected flow networks. The algorithm also imposes two constraints on the directed flow networks, namely, is doesn't support infinite capacity arcs with negative cost and self-loops. Note, that in the case the network contains an infinite capacity arc with negative cost, the cost of a flow on the network can be bounded from below by some constant, i.e. a feasible finite weight solution can exist.

    An arc with capacity greater that or equal to CAP_INF is considered to be an infinite capacity arc. The algorithm also uses COST_INF during the computation, therefore, the magnitude of the cost of any arc can't exceed this values.

    In the capacity scaling mode, the algorithm performs $\mathcal{O}(log_a U)$ $\Delta$-scaling phases, where $U$ is the largest magnitude of any supply/demand or finite arc capacity, and $a$ is a scaling factor, which is considered to be constant. During each $\Delta$-scaling phase the algorithm first ensures that all arc with capacity with capacity greater than or equal to $\Delta$ satisfy optimality condition, i.e. its reduced cost must be non-negative (saturated arcs don't belong to the residual network). After saturating all arcs in the $\Delta$-residual network with negative reduced cost the sum of the excesses is bounded by $2\Delta(m + n)$. Since the algorithm ensures that each augmentation carries at least $\Delta$ units of flow, at most $\mathcal{O}(m)$ flow augmentations are performed during each scaling phase. Therefore, the overall running time of the algorithm with capacity scaling is $\mathcal{O}(m\log_a U(m + n)\log n)$, which is a weakly polynomial time bound.

    If the algorithm is used without scaling, each flow augmentation carries at least $\mathcal{O}(1)$ flow units, therefore the overall time complexity if $\mathcal{O}(nU(m + n)\log n)$, which is a pseudo-polynomial time bound.

    For more information about the capacity scaling algorithm see: K. Ahuja, Ravindra & L. Magnanti, Thomas & Orlin, James. (1993). Network Flows. This implementation is based on the algorithm description presented in this book.

    Timofey Chudakov
    See Also:
    MinimumCostFlowProblem, MinimumCostFlowAlgorithm
    • Field Detail

      • CAP_INF

        public static final int CAP_INF
        A capacity which is considered to be infinite. Every arc, which has upper capacity greater that or equal to this value is considered to be an infinite capacity arc.
        See Also:
        Constant Field Values
      • COST_INF

        public static final double COST_INF
        A cost which is considered to be infinite. This value is used internally for flow network transformation. That is why arcs with cost magnitude greater than or equal to this value are not allowed.
        See Also:
        Constant Field Values

        public static final int DEFAULT_SCALING_FACTOR
        Default scaling factor
        See Also:
        Constant Field Values
    • Constructor Detail

      • CapacityScalingMinimumCostFlow

        public CapacityScalingMinimumCostFlow()
        Constructs a new instance of the algorithm which uses default scaling factor.
      • CapacityScalingMinimumCostFlow

        public CapacityScalingMinimumCostFlow​(int scalingFactor)
        Constructs a new instance of the algorithm with custom scalingFactor. If the scalingFactor is less than 2, the algorithm doesn't use scaling.
        scalingFactor - custom scaling factor
    • Method Detail

      • getFlowMap

        public java.util.Map<E,​java.lang.Double> getFlowMap()
        Returns mapping from edge to flow value through this particular edge
        Specified by:
        getFlowMap in interface FlowAlgorithm<V,​E>
        maximum flow mapping, or null if a MinimumCostFlowProblem has not yet been solved.
      • getFlowDirection

        public V getFlowDirection​(E edge)
        For the specified edge $(u, v)$ returns vertex $v$ if the flow goes from $u$ to $v$, or returns vertex $u$ otherwise. For directed flow networks the result is always the head of the specified arc.

        Note: not all flow algorithms may support undirected graphs.

        Specified by:
        getFlowDirection in interface FlowAlgorithm<V,​E>
        edge - an edge from the specified flow network
        the direction of the flow on the edge
      • getDualSolution

        public java.util.Map<V,​java.lang.Double> getDualSolution()
        Returns solution to the dual linear program formulated on the network. Serves as a certificate of optimality.

        It is represented as a mapping from graph nodes to their potentials (dual variables). Reduced cost of a arc $(a, b)$ is defined as $cost((a, b)) + potential(b) - potential(b)$. According to the reduced cost optimality conditions, a feasible solution to the minimum cost flow problem is optimal if and only if reduced cost of every non-saturated arc is greater than or equal to $0$.

        solution to the dual linear program formulated on the network, or null if a MinimumCostFlowProblem has not yet been solved.
      • testOptimality

        public boolean testOptimality​(double eps)
        Tests the optimality conditions after a flow of minimum cost has been computed.

        More precisely, tests, whether the reduced cost of every non-saturated arc in the residual network is non-negative. This validation is performed with precision of eps. If the solution doesn't meet this condition, returns, false.

        In general, this method should always return true unless the algorithm implementation has a bug.

        eps - the precision to use
        true, if the computed solution is optimal, false otherwise.