Class KolmogorovWeightedPerfectMatching<V,E>
- Type Parameters:
V
- the graph vertex typeE
- the graph edge type
- All Implemented Interfaces:
MatchingAlgorithm<V,
E>
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:
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Nested Class Summary
Nested ClassesModifier and TypeClassDescriptionstatic class
A solution to the dual linear program formulated on thegraph
static class
Describes the performance characteristics of the algorithm and numeric data about the number of performed dual operations during the main phase of the algorithmNested classes/interfaces inherited from interface org.jgrapht.alg.interfaces.MatchingAlgorithm
MatchingAlgorithm.Matching<V,
E>, MatchingAlgorithm.MatchingImpl<V, E> -
Field Summary
FieldsModifier and TypeFieldDescription(package private) static final boolean
When set to true, verbose debugging output will be producedstatic final BlossomVOptions
Default optionsDefines solution to the dual linear program formulated on thegraph
private BlossomVDualUpdater
<V, E> Performs dual updates using the strategy defined by theoptions
static final double
Default epsilon used in the algorithmThe graph we are matching onstatic final double
Default infinity value used in the algorithmInitial graph specified during the construction timeprivate MatchingAlgorithm.Matching
<V, E> The computed matching of thegraph
(package private) static final String
Exception message if no perfect matching is possiblestatic final double
Defines the threshold for throwing an exception about no perfect matching existenceprivate ObjectiveSense
The objective sense of the algorithm, i.e.private BlossomVOptions
BlossomVOptions used by the algorithm to match the problem instanceprivate BlossomVPrimalUpdater
<V, E> Performs primal operations (grow, augment, shrink and expand)(package private) BlossomVState
<V, E> Current state of the algorithmFields inherited from interface org.jgrapht.alg.interfaces.MatchingAlgorithm
DEFAULT_EPSILON
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Constructor Summary
ConstructorsConstructorDescriptionKolmogorovWeightedPerfectMatching
(Graph<V, E> graph) Constructs a new instance of the algorithm using the default options.KolmogorovWeightedPerfectMatching
(Graph<V, E> graph, BlossomVOptions options) Constructs a new instance of the algorithm with the specifiedoptions
.KolmogorovWeightedPerfectMatching
(Graph<V, E> graph, BlossomVOptions options, ObjectiveSense objectiveSense) Constructs a new instance of the algorithm with the specifiedoptions
.KolmogorovWeightedPerfectMatching
(Graph<V, E> graph, ObjectiveSense objectiveSense) Constructs a new instance of the algorithm using the default options. -
Method Summary
Modifier and TypeMethodDescriptionprivate void
Clears the marking of all nodes and pseudonodesprivate void
clearMarked
(BlossomVNode node) Clears the marking ofnode
and all its ancestors up until the first unmarked vertex is encounteredprivate void
finish()
Finishes the algorithm after all nodes are matched.getBlossomNodes
(BlossomVNode pseudonode, Map<BlossomVNode, Set<V>> blossomNodes) Computes the set of original contracted vertices in thepseudonode
and puts computes value into theblossomNodes
.Returns the computed solution to the dual linear program with respect to the weighted perfect matching linear program formulation.double
getError()
Computes the error in the solution to the dual linear program.Computes and returns a weighted perfect matching in thegraph
.Returns the statistics describing the performance characteristics of the algorithm.Computes a solution to a dual linear program formulated on the initial graph.private void
Lazily runs the algorithm on the specified graph.private Pair
<BlossomVNode, BlossomVNode> lca
(BlossomVNode a, BlossomVNode b) Returns $(b, b)$ in the case where the verticesa
andb
have a common ancestor blossom $b$.private void
Sets the blossomGrandparent references so that from a pseudonode we can make one step down to some node that belongs to that pseudonodeprivate void
printMap()
Debug methodprivate void
Prints the state of the algorithm.private void
Debug methodprivate void
Sets the currentEdge and currentDirection variables for all trees adjacent to thetree
private double
Tests whether a non-negative dual variable is assigned to every blossomboolean
Performs an optimality test after the perfect matching is computed.private double
totalDual
(BlossomVNode start, BlossomVNode end) Computes the sum of all duals fromstart
inclusive toend
inclusive
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Field Details
-
EPS
public static final double EPSDefault epsilon used in the algorithm- See Also:
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INFINITY
public static final double INFINITYDefault infinity value used in the algorithm- See Also:
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NO_PERFECT_MATCHING_THRESHOLD
public static final double NO_PERFECT_MATCHING_THRESHOLDDefines the threshold for throwing an exception about no perfect matching existence- See Also:
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DEFAULT_OPTIONS
Default options -
DEBUG
static final boolean DEBUGWhen set to true, verbose debugging output will be produced- See Also:
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NO_PERFECT_MATCHING
Exception message if no perfect matching is possible- See Also:
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initialGraph
Initial graph specified during the construction time -
graph
The graph we are matching on -
state
BlossomVState<V,E> stateCurrent state of the algorithm -
primalUpdater
Performs primal operations (grow, augment, shrink and expand) -
dualUpdater
Performs dual updates using the strategy defined by theoptions
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matching
The computed matching of thegraph
-
dualSolution
Defines solution to the dual linear program formulated on thegraph
-
options
BlossomVOptions used by the algorithm to match the problem instance -
objectiveSense
The objective sense of the algorithm, i.e. whether to maximize or minimize the weight of the resulting perfect matching
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Constructor Details
-
KolmogorovWeightedPerfectMatching
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
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KolmogorovWeightedPerfectMatching
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 themaximize
parameter.- Parameters:
graph
- the graph for which to find a weighted perfect matchingobjectiveSense
- objective sense of the algorithm
-
KolmogorovWeightedPerfectMatching
Constructs a new instance of the algorithm with the specifiedoptions
. 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 matchingoptions
- the options which define the strategies for the initialization and dual updates
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KolmogorovWeightedPerfectMatching
public KolmogorovWeightedPerfectMatching(Graph<V, E> graph, BlossomVOptions options, ObjectiveSense objectiveSense) Constructs a new instance of the algorithm with the specifiedoptions
. The goal of the constructed algorithm is to maximize or minimize the weight of the resulting perfect matching depending on themaximize
parameter.- Parameters:
graph
- the graph for which to find a weighted perfect matchingoptions
- the options which define the strategies for the initialization and dual updatesobjectiveSense
- objective sense of the algorithm
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Method Details
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getMatching
Computes and returns a weighted perfect matching in thegraph
. See the class description for the relative definitions and algorithm description.- Specified by:
getMatching
in interfaceMatchingAlgorithm<V,
E> - Returns:
- a weighted perfect matching for the
graph
-
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
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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
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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
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lazyComputeWeightedPerfectMatching
private void lazyComputeWeightedPerfectMatching()Lazily runs the algorithm on the specified graph. -
setCurrentEdgesAndTryToAugment
Sets the currentEdge and currentDirection variables for all trees adjacent to thetree
- Parameters:
tree
- the tree whose adjacent trees' variables are modified
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testNonNegativity
private double testNonNegativity()Tests whether a non-negative dual variable is assigned to every blossom- Returns:
- true iff the condition described above holds
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totalDual
Computes the sum of all duals fromstart
inclusive toend
inclusive- Parameters:
start
- the node to start fromend
- the node to end with- Returns:
- the sum = start.dual + start.blossomParent.dual + ... + end.dual
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lca
Returns $(b, b)$ in the case where the verticesa
andb
have a common ancestor blossom $b$. Otherwise, returns the outermost parent blossoms of nodesa
andb
- Parameters:
a
- a vertex whose lca is to be found with respect to another vertexb
- the other vertex whose lca is to be found- Returns:
- either an lca blossom of
a
andb
or their outermost blossoms
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clearMarked
Clears the marking ofnode
and all its ancestors up until the first unmarked vertex is encountered- Parameters:
node
- the node to start from
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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.
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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
Computes the set of original contracted vertices in thepseudonode
and puts computes value into theblossomNodes
. Ifnode
contains other pseudonodes which haven't been processed already, recursively computes the same set for them.- Parameters:
pseudonode
- the pseudonode whose contracted nodes are computedblossomNodes
- the mapping from pseudonodes to the original nodes contained in them
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lazyComputeDualSolution
Computes a solution to a dual linear program formulated on the initial graph.- Returns:
- the solution to the dual linear program
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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
Returns the statistics describing the performance characteristics of the algorithm.- Returns:
- the statistics describing the algorithms characteristics
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