org.jgrapht.alg.matching.blossom.v5

## Class KolmogorovWeightedPerfectMatching<V,E>

• java.lang.Object
• org.jgrapht.alg.matching.blossom.v5.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 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, EdmondsMaximumCardinalityMatching 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.

Author:
Timofey Chudakov
KolmogorovWeightedMatching, BlossomVPrimalUpdater, BlossomVDualUpdater
• ### Nested Class Summary

Nested Classes
Modifier and Type Class and Description
static class  KolmogorovWeightedPerfectMatching.DualSolution<V,E>
A solution to the dual linear program formulated on the graph
static class  KolmogorovWeightedPerfectMatching.Statistics
Describes the performance characteristics of the algorithm and numeric data about the number of performed dual operations during the main phase of the algorithm
• ### Nested classes/interfaces inherited from interface org.jgrapht.alg.interfaces.MatchingAlgorithm

MatchingAlgorithm.Matching<V,E>, MatchingAlgorithm.MatchingImpl<V,E>
• ### Field Summary

Fields
Modifier and Type Field and Description
static BlossomVOptions DEFAULT_OPTIONS
Default options
static double EPS
Default epsilon used in the algorithm
static double INFINITY
Default infinity value used in the algorithm
static double NO_PERFECT_MATCHING_THRESHOLD
Defines the threshold for throwing an exception about no perfect matching existence
• ### Fields inherited from interface org.jgrapht.alg.interfaces.MatchingAlgorithm

DEFAULT_EPSILON
• ### Constructor Summary

Constructors
Constructor and Description
KolmogorovWeightedPerfectMatching(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 specified options.
KolmogorovWeightedPerfectMatching(Graph<V,E> graph, BlossomVOptions options, ObjectiveSense objectiveSense)
Constructs a new instance of the algorithm with the specified options.
KolmogorovWeightedPerfectMatching(Graph<V,E> graph, ObjectiveSense objectiveSense)
Constructs a new instance of the algorithm using the default options.
• ### Method Summary

All Methods
Modifier and Type Method and Description
KolmogorovWeightedPerfectMatching.DualSolution<V,E> getDualSolution()
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.
MatchingAlgorithm.Matching<V,E> getMatching()
Computes and returns a weighted perfect matching in the graph.
KolmogorovWeightedPerfectMatching.Statistics getStatistics()
Returns the statistics describing the performance characteristics of the algorithm.
boolean testOptimality()
Performs an optimality test after the perfect matching is computed.
• ### Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
• ### Field Detail

• #### EPS

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

public static final double INFINITY
Default infinity value used in the algorithm
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
Constant Field Values
• #### DEFAULT_OPTIONS

public static final BlossomVOptions DEFAULT_OPTIONS
Default options
• ### 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
• #### getStatistics

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