Class KolmogorovWeightedPerfectMatching<V,​E>

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

    public class KolmogorovWeightedPerfectMatching<V,​E>
    extends java.lang.Object
    implements MatchingAlgorithm<V,​E>
    This class computes weighted perfect matchings in general graphs using the Blossom V algorithm. If maximum or minimum weight matching algorithms is needed, see KolmogorovWeightedMatching

    Let $G = (V, E, c)$ be an undirected graph with a real-valued cost function defined on it. A matching is an edge-disjoint subset of edges $M \subseteq E$. A matching is perfect if $2|M| = |V|$. In the weighted perfect matching problem the goal is to maximize or minimize the weighted sum of the edges in the matching. This class supports pseudographs, but a problem on a pseudograph can be easily reduced to a problem on a simple graph. Moreover, this reduction can heavily influence the running time since only an edge with a maximum or minimum weight between two vertices can belong to the matching in the corresponding optimization problems. Currently, users are responsible for doing this reduction themselves before invoking the algorithm.

    Note that if the graph is unweighted and dense, SparseEdmondsMaximumCardinalityMatching may be a better choice.

    For more information about the algorithm see the following paper: Kolmogorov, V. Math. Prog. Comp. (2009) 1: 43. https://doi.org/10.1007/s12532-009-0002-8, and the original implementation: http://pub.ist.ac.at/~vnk/software/blossom5-v2.05.src.tar.gz

    The algorithm can be divided into two phases: initialization and the main algorithm. The initialization phase is responsible for converting the specified graph into the form convenient for the algorithm and for finding an initial matching to speed up the main part. Furthermore, the main part of the algorithm can be further divided into primal and dual updates. The primal phases are aimed at augmenting the matching so that the value of the objective function of the primal linear program increases. Dual updates are aimed at increasing the objective function of the dual linear program. The algorithm iteratively performs these primal and dual operations to build alternating trees of tight edges and augment the matching. Thus, at any stage of the algorithm the matching consists of tight edges. This means that the resulting perfect matching meets complementary slackness conditions, and is therefore optimal.

    At construction time the set of options can be specified to define the strategies used by the algorithm to perform initialization, dual updates, etc. This can be done with the BlossomVOptions. During the construction time the objective sense of the optimization problem can be specified, i.e. whether to maximize of minimize the weight of the resulting perfect matching. Default objective sense of the algorithm is to minimize the weight of the resulting perfect matching. If the objective sense of the algorithm is to maximize the weight of the matching, the problem is reduced to minimum weight perfect matching problem by multiplying all edge weights by $-1$. This class supports retrieving statistics for the algorithm performance, see getStatistics(). It provides the time elapsed during primal operations and dual updates, as well as the number of these primal operations performed.

    The solution to a weighted perfect matching problem instance comes with a certificate of optimality, which is represented by a solution to a dual linear program; see KolmogorovWeightedPerfectMatching.DualSolution. This class encapsulates a mapping from the node sets of odd cardinality to the corresponding dual variables. This mapping doesn't contain the sets whose dual variables are $0$. The computation of the dual solution is performed lazily and doesn't affect the running time of finding a weighted perfect matching.

    Here we describe the certificates of optimality more precisely. Let the graph $G = (V, E)$ be an undirected graph with cost function $c: V \mapsto \mathbb{R}$ defined on it. Let $\mathcal{O}$ be the set of all subsets of $V$ of odd cardinality containing at least 3 vertices, and $\delta(S), S \subset V$ be the set of boundary edges of $V$. Then minimum weight perfect matching problem has the following linear programming formulation: \[ \begin{align} \mbox{minimize} \qquad & \sum_{e\in E}c_e \cdot x_e &\\ s.t. \qquad & \sum_{e\in \delta^(i)} x_e = 1 & \forall i\in V\\ & \sum_{e\in \delta(S)}x_e \ge 1 & \forall S\in \mathcal{O} \\ & x_e \ge 0 & \forall e\in E \end{align}\] The corresponding dual linear program has the following form: \[ \begin{align} \mbox{maximize} \qquad & \sum_{x \in V}y_e &\\ s.t. \qquad & y_u + y_v + \sum_{S\in \mathcal{O}: e \in \delta(S)}y_S \le c_e & \forall\ e = \{u, v\}\in E\\ & x_S \ge 0 & \forall S\in \mathcal{O} \end{align} \] Let's use the following notation: $slack(e) = c_e - y_u - y_v - \sum_{S\in \mathcal{O}: e \in \delta(S)}y_S$. Complementary slackness conditions have the following form: \[ \begin{align} slack(e) > 0 &\Rightarrow x_e = 0 \\ y_S > 0 &\Rightarrow \sum_{e\in \delta(S)}x_e = 1 \end{align} \] Therefore, the slacks of all edges will be non-negative and the slacks of matched edges will be $0$.

    The maximum weight perfect matching problem has the following linear programming formulation: \[ \begin{align} \mbox{maximize} \qquad & \sum_{e\in E}c_e \cdot x_e &\\ s.t. \qquad &\sum_{e\in \delta^(i)} x_e = 1 & \forall i\in V\\ & \sum_{e\in \delta(S)}x_e \ge 1 & \forall S\in \mathcal{O} \\ & x_e \ge 0 & \forall e\in E \end{align} \] The corresponding dual linear program has the following form: \[ \begin{align} \mbox{minimize} \qquad & \sum_{x \in V}y_e &\\ s.t. \qquad & y_u + y_v + \sum_{S\in \mathcal{O}: e \in \delta(S)}y_S \ge c_e & \forall\ e = \{u, v\}\in E\\ & x_S \le 0 & \forall S\in \mathcal{O} \end{align} \] Complementary slackness conditions have the following form: \[ \begin{align} slack(e) < 0 &\Rightarrow x_e = 0 \\ y_S < 0 &\Rightarrow \sum_{e\in \delta(S)}x_e = 1 \end{align} \] Therefore, the slacks of all edges will be non-positive and the slacks of matched edges will be $0$.

    This class supports testing the optimality of the solution via testOptimality(). It also supports retrieval of the computation error when the edge weights are real values via getError(). Both optimality test and error computation are performed lazily and don't affect the running time of the main algorithm. If the problem instance doesn't contain a perfect matching at all, the algorithm doesn't find a minimum weight maximum matching; instead, it throws an exception.

    See Also:
    KolmogorovWeightedMatching, BlossomVPrimalUpdater, BlossomVDualUpdater
    • Field Detail

      • EPS

        public static final double EPS
        Default epsilon used in the algorithm
        See Also:
        Constant Field Values
      • INFINITY

        public static final double INFINITY
        Default infinity value used in the algorithm
        See Also:
        Constant Field Values
      • NO_PERFECT_MATCHING_THRESHOLD

        public static final double NO_PERFECT_MATCHING_THRESHOLD
        Defines the threshold for throwing an exception about no perfect matching existence
        See Also:
        Constant Field Values
      • DEFAULT_OPTIONS

        public static final BlossomVOptions DEFAULT_OPTIONS
        Default options
      • DEBUG

        static final boolean DEBUG
        When set to true, verbose debugging output will be produced
        See Also:
        Constant Field Values
      • NO_PERFECT_MATCHING

        static final java.lang.String NO_PERFECT_MATCHING
        Exception message if no perfect matching is possible
        See Also:
        Constant Field Values
      • initialGraph

        private final Graph<V,​E> initialGraph
        Initial graph specified during the construction time
      • graph

        private final Graph<V,​E> graph
        The graph we are matching on
      • primalUpdater

        private BlossomVPrimalUpdater<V,​E> primalUpdater
        Performs primal operations (grow, augment, shrink and expand)
      • dualUpdater

        private BlossomVDualUpdater<V,​E> dualUpdater
        Performs dual updates using the strategy defined by the options
      • options

        private BlossomVOptions options
        BlossomVOptions used by the algorithm to match the problem instance
      • objectiveSense

        private ObjectiveSense objectiveSense
        The objective sense of the algorithm, i.e. whether to maximize or minimize the weight of the resulting perfect matching
    • Constructor Detail

      • KolmogorovWeightedPerfectMatching

        public KolmogorovWeightedPerfectMatching​(Graph<V,​E> graph)
        Constructs a new instance of the algorithm using the default options. The goal of the constructed algorithm is to minimize the weight of the resulting perfect matching.
        Parameters:
        graph - the graph for which to find a weighted perfect matching
      • KolmogorovWeightedPerfectMatching

        public KolmogorovWeightedPerfectMatching​(Graph<V,​E> graph,
                                                 ObjectiveSense objectiveSense)
        Constructs a new instance of the algorithm using the default options. The goal of the constructed algorithm is to maximize or minimize the weight of the resulting perfect matching depending on the maximize parameter.
        Parameters:
        graph - the graph for which to find a weighted perfect matching
        objectiveSense - objective sense of the algorithm
      • KolmogorovWeightedPerfectMatching

        public KolmogorovWeightedPerfectMatching​(Graph<V,​E> graph,
                                                 BlossomVOptions options)
        Constructs a new instance of the algorithm with the specified options. The objective sense of the constructed algorithm is to minimize the weight of the resulting matching
        Parameters:
        graph - the graph for which to find a weighted perfect matching
        options - the options which define the strategies for the initialization and dual updates
      • KolmogorovWeightedPerfectMatching

        public KolmogorovWeightedPerfectMatching​(Graph<V,​E> graph,
                                                 BlossomVOptions options,
                                                 ObjectiveSense objectiveSense)
        Constructs a new instance of the algorithm with the specified options. The goal of the constructed algorithm is to maximize or minimize the weight of the resulting perfect matching depending on the maximize parameter.
        Parameters:
        graph - the graph for which to find a weighted perfect matching
        options - the options which define the strategies for the initialization and dual updates
        objectiveSense - objective sense of the algorithm
    • Method Detail

      • getMatching

        public MatchingAlgorithm.Matching<V,​E> getMatching()
        Computes and returns a weighted perfect matching in the graph. See the class description for the relative definitions and algorithm description.
        Specified by:
        getMatching in interface MatchingAlgorithm<V,​E>
        Returns:
        a weighted perfect matching for the graph
      • getDualSolution

        public KolmogorovWeightedPerfectMatching.DualSolution<V,​E> getDualSolution()
        Returns the computed solution to the dual linear program with respect to the weighted perfect matching linear program formulation.
        Returns:
        the solution to the dual linear program formulated on the graph
      • testOptimality

        public boolean testOptimality()
        Performs an optimality test after the perfect matching is computed.

        More precisely, checks whether dual variables of all pseudonodes and resulting slacks of all edges are non-negative and that slacks of all matched edges are exactly 0. Since the algorithm uses floating point arithmetic, this check is done with precision of EPS

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

        Returns:
        true iff the assigned dual variables satisfy the dual linear program formulation AND complementary slackness conditions are also satisfied. The total error must not exceed EPS
      • getError

        public double getError()
        Computes the error in the solution to the dual linear program. More precisely, the total error equals the sum of:
        • Absolute value of edge slack if negative or the edge is matched
        • Absolute value of pseudonode variable if negative
        Returns:
        the total numeric error
      • lazyComputeWeightedPerfectMatching

        private void lazyComputeWeightedPerfectMatching()
        Lazily runs the algorithm on the specified graph.
      • setCurrentEdgesAndTryToAugment

        private void setCurrentEdgesAndTryToAugment​(BlossomVTree tree)
        Sets the currentEdge and currentDirection variables for all trees adjacent to the tree
        Parameters:
        tree - the tree whose adjacent trees' variables are modified
      • testNonNegativity

        private double testNonNegativity()
        Tests whether a non-negative dual variable is assigned to every blossom
        Returns:
        true iff the condition described above holds
      • totalDual

        private double totalDual​(BlossomVNode start,
                                 BlossomVNode end)
        Computes the sum of all duals from start inclusive to end inclusive
        Parameters:
        start - the node to start from
        end - the node to end with
        Returns:
        the sum = start.dual + start.blossomParent.dual + ... + end.dual
      • lca

        private Pair<BlossomVNode,​BlossomVNode> lca​(BlossomVNode a,
                                                          BlossomVNode b)
        Returns $(b, b)$ in the case where the vertices a and b have a common ancestor blossom $b$. Otherwise, returns the outermost parent blossoms of nodes a and b
        Parameters:
        a - a vertex whose lca is to be found with respect to another vertex
        b - the other vertex whose lca is to be found
        Returns:
        either an lca blossom of a and b or their outermost blossoms
      • clearMarked

        private void clearMarked​(BlossomVNode node)
        Clears the marking of node and all its ancestors up until the first unmarked vertex is encountered
        Parameters:
        node - the node to start from
      • clearMarked

        private void clearMarked()
        Clears the marking of all nodes and pseudonodes
      • finish

        private void finish()
        Finishes the algorithm after all nodes are matched. The main problem it solves is that the matching after the end of primal and dual operations may not be valid in the contracted blossoms.

        Property: if a matching is changed in the parent blossom, the matching in all lower blossoms can become invalid. Therefore, we traverse all nodes, find an unmatched node (it is necessarily contracted), go up to the first blossom whose matching hasn't been fixed (we set blossomGrandparent references to point to the previous nodes on the path). Then we start to change the matching accordingly all the way down to the initial node.

        Let's call an edge that is matched to a blossom root a "blossom edge". To make the matching valid we move the blossom edge one layer down at a time so that in the end its endpoints are valid initial nodes of the graph. After this transformation we can't traverse the blossomSibling references any more. That is why we initially compute a mapping of every pseudonode to the set of nodes that are contracted in it. This map is needed to construct a dual solution after the matching in the graph becomes valid.

      • prepareForDualSolution

        private void prepareForDualSolution()
        Sets the blossomGrandparent references so that from a pseudonode we can make one step down to some node that belongs to that pseudonode
      • getBlossomNodes

        private java.util.Set<V> getBlossomNodes​(BlossomVNode pseudonode,
                                                 java.util.Map<BlossomVNode,​java.util.Set<V>> blossomNodes)
        Computes the set of original contracted vertices in the pseudonode and puts computes value into the blossomNodes. If node contains other pseudonodes which haven't been processed already, recursively computes the same set for them.
        Parameters:
        pseudonode - the pseudonode whose contracted nodes are computed
        blossomNodes - the mapping from pseudonodes to the original nodes contained in them
      • lazyComputeDualSolution

        private KolmogorovWeightedPerfectMatching.DualSolution<V,​E> lazyComputeDualSolution()
        Computes a solution to a dual linear program formulated on the initial graph.
        Returns:
        the solution to the dual linear program
      • printState

        private void printState()
        Prints the state of the algorithm. This is a debug method.
      • printTrees

        private void printTrees()
        Debug method
      • printMap

        private void printMap()
        Debug method
      • getStatistics

        public KolmogorovWeightedPerfectMatching.Statistics getStatistics()
        Returns the statistics describing the performance characteristics of the algorithm.
        Returns:
        the statistics describing the algorithms characteristics