Class CapacityScalingMinimumCostFlow<V,E>

java.lang.Object
org.jgrapht.alg.flow.mincost.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 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.

See Also:

  • Field Details

    • 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:

    • 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:

    • DEFAULT_SCALING_FACTOR

      public static final int DEFAULT_SCALING_FACTOR
      Default scaling factor
      See Also:

    • DEBUG

      private static final boolean DEBUG
      Debug variable
      See Also:

    • scalingFactor

      private final int scalingFactor
      Scaling factor of this algorithm

    • n

      private int n
      Number of vertices in the network

    • m

      private int m
      Number of edges in the network

    • counter

      private int counter
      Variable that is used to determine whether a vertex has been labeled temporarily or permanently during Dijkstra's algorithm

    • problem

      private MinimumCostFlowProblem<V,E> problem
      Specified minimum cost flow problem

    • minimumCostFlow

      private MinimumCostFlowAlgorithm.MinimumCostFlow<E> minimumCostFlow
      Computed minimum cost flow

    • nodes

      private CapacityScalingMinimumCostFlow.Node[] nodes
      Array of internal nodes used by the algorithm. Node: these nodes are stored in the same order as vertices of the specified flow network. This allows to determine quickly their counterparts in the network.

    • arcs

      private CapacityScalingMinimumCostFlow.Arc[] arcs
      Array of internal arcs. Note: these arcs are stored in the same order as edges of the specified flow network. This allows to determine quickly their counterparts in the network.

    • graphVertices

      private List<V> graphVertices
      List of vertices of the flow network.

    • graphEdges

      private List<E> graphEdges
      List of edges of the flow network.

  • Constructor Details

    • 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.
      Parameters:
      scalingFactor - custom scaling factor

  • Method Details

    • getFlowMap

      public Map<E,Double> getFlowMap()
      Returns mapping from edge to flow value through this particular edge
      Specified by:
      getFlowMap in interface FlowAlgorithm<V,E>
      Returns:
      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>
      Parameters:
      edge - an edge from the specified flow network
      Returns:
      the direction of the flow on the edge

    • getMinimumCostFlow

      public MinimumCostFlowAlgorithm.MinimumCostFlow<E> getMinimumCostFlow(MinimumCostFlowProblem<V,E> minimumCostFlowProblem)
      Calculates feasible flow of minimum cost for the minimum cost flow problem.
      Specified by:
      getMinimumCostFlow in interface MinimumCostFlowAlgorithm<V,E>
      Parameters:
      minimumCostFlowProblem - minimum cost flow problem
      Returns:
      minimum cost flow

    • getDualSolution

      public Map<V,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 $0$.

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

    • calculateMinimumCostFlow

      private void calculateMinimumCostFlow()
      Calculated a solution to the specified minimum cost flow problem. If the scaling factor is greater than 1, performs scaling phases, otherwise uses simple capacity scaling algorithm.

    • init

      private void init()
      Converts the flow network in the form convenient for the algorithm. Validated the arc capacities and costs.

      Also, adds a dummy node to the network and arcs from every node to this dummy node, and from this dummy node to every other node. These added arcs have infinite capacities CAP_INF and infinite costs COST_INF. This ensures, that every search for an augmenting path to send at least $\Delta$ units of flow succeeds.

      If the flow network has a feasible solution, at the end there will be no flow on the added arcs. Otherwise, the specified problem has no feasible solution.

    • getU

      private int getU()
      Returns the largest magnitude of any supply/demand or finite arc capacity.
      Returns:
      the largest magnitude of any supply/demand or finite arc capacity.

    • scale

      private Pair<List<CapacityScalingMinimumCostFlow.Node>, Set<CapacityScalingMinimumCostFlow.Node>> scale(int delta)
      Performs a scaling phase by saturating all negative reduced cost arcs with residual capacity greater than or equal to the delta, so that they don't belong to the $\Delta$-residual network and, hence, don't violate optimality conditions. After that this method computes and returns nodes with positive excess greater than or equal to the delta and nodes with negative excesses that are less than or equal to delta
      Parameters:
      delta - current value of $\Delta$
      Returns:
      the nodes with excesses no less than delta and no greater than -delta

    • pushAllFlow

      private void pushAllFlow(List<CapacityScalingMinimumCostFlow.Node> positiveExcessNodes, Set<CapacityScalingMinimumCostFlow.Node> negativeExcessNodes, int delta)
      For every node in the positiveExcessNodes pushes all flow away from it until its excess is less than delta. This is always possible due to the performed flow network reduction during the initialization phase.
      Parameters:
      positiveExcessNodes - nodes from the network with positive excesses no less than delta
      negativeExcessNodes - nodes from the network with negative excesses no greater than delta
      delta - the current value of $\Delta$

    • pushDijkstra

      private void pushDijkstra(CapacityScalingMinimumCostFlow.Node start, Set<CapacityScalingMinimumCostFlow.Node> negativeExcessNodes, int delta)
      Runs the Dijkstra's algorithm in the residual network using CapacityScalingMinimumCostFlow.Arc.getReducedCost() as arc distances.

      After reaching a node with excess no greater than -delta, augments it. Since the search is performed in the $\Delta$-residual network, the augmentation carries at least delta units of flow. The search always succeeds due to the flow network reduction performed during the initialization phase.

      Updates the potentials of the nodes so that they:

      • Satisfy optimality conditions in the $\Delta$-residual network
      • The reduced cost of the augmented path is equal to $0$

      Let us denote some permanently labeled vertex as $u$, and the first permanently labeled vertex with negative excess as $v$. Let $dist(x)$ be the distance function in the residual network. Then we use the following formula to update the node potentials: $v.potential = v.potential + dist(v) - dist(u)$. The potentials of the temporarily labeled and unvisited vertices stay unchanged.

      Parameters:
      start - the start node for Dijkstra's algorithm
      negativeExcessNodes - nodes from the network with negative excesses no greater than delta
      delta - the current value of $\Delta$

    • augmentPath

      private void augmentPath(CapacityScalingMinimumCostFlow.Node start, CapacityScalingMinimumCostFlow.Node end)
      Augments the path from start to the end sending as much flow as possible. Uses CapacityScalingMinimumCostFlow.Node.parentArc computed by the Dijkstra's algorithm. Updates the excesses of the start and the end nodes.
      Parameters:
      start - the start of the augmenting path
      end - the end of the augmenting path

    • finish

      private MinimumCostFlowAlgorithm.MinimumCostFlow<E> finish()
      Finishes the computation by checking the flow feasibility, computing arc flows, and creating an instance of MinimumCostFlowAlgorithm.MinimumCostFlow. The resulting flow mapping contains all edges of the specified minimum cost flow problem.
      Returns:
      the solution to the minimum cost flow problem

    • 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.

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