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/*
* (C) Copyright 2003-2021, by Joris Kinable and Contributors.
*
* JGraphT : a free Java graph-theory library
*
* See the CONTRIBUTORS.md file distributed with this work for additional
* information regarding copyright ownership.
*
* This program and the accompanying materials are made available under the
* terms of the Eclipse Public License 2.0 which is available at
* http://www.eclipse.org/legal/epl-2.0, or the
* GNU Lesser General Public License v2.1 or later
* which is available at
* http://www.gnu.org/licenses/old-licenses/lgpl-2.1-standalone.html.
*
* SPDX-License-Identifier: EPL-2.0 OR LGPL-2.1-or-later
*/
package org.jgrapht.alg.vertexcover;
import org.jgrapht.*;
import org.jgrapht.alg.interfaces.*;
import org.jgrapht.alg.util.*;
import java.util.*;
import java.util.function.*;
import java.util.stream.*;
/**
* Finds a minimum vertex cover in a undirected graph. The implementation relies on a recursive
* algorithm. At each recursive step, the algorithm picks a unvisited vertex v and distinguishes two
* cases: either v has to be added to the vertex cover or all of its neighbors.
*
* In pseudo code, the algorithm (simplified) looks like this:
*
*
*
* $VC(G)$:
* if $V = \emptyset$ then return $\emptyset$
* Choose an arbitrary node $v \in G$
* $G1 := (V − v, \left{ e \in E | v \not \in e \right})$
* $G2 := (V − v − N(v), \left{ e \in E | e \cap (N(v) \cup v)= \empty \right})$
* if $|v \cup VC(G1)| \leq |N(v) \cup VC(G2)|$ then
* return $v \cup VC(G1)$
* else
* return $N(v) \cup VC(G2)$
*
*
*
* To speed up the implementation, memoization and a bounding procedure are used. The current
* implementation solves instances with 150-250 vertices efficiently to optimality.
*
* TODO JK: determine runtime complexity and add it to class description. TODO JK: run this class
* through a performance profiler
*
* @param the graph vertex type
* @param the graph edge type
*
* @author Joris Kinable
*/
public class RecursiveExactVCImpl
implements
VertexCoverAlgorithm
{
/** Input graph **/
private Graph graph;
/** Number of vertices in the graph **/
private int n;
/**
* Neighbor cache TODO JK: It might be worth trying to replace the neighbors index by a bitset
* view. As such, all operations can be simplified to bitset operations, which may improve the
* algorithm's performance.
**/
private NeighborCache neighborCache;
/** Map for memoization **/
private Map memo;
/**
* Ordered list of vertices which will be iteratively considered to be included in a matching
**/
private List vertices;
/** Mapping of a vertex to its index in the list of vertices **/
private Map vertexIDDictionary;
/**
* Maximum weight of the vertex cover. In case there is no weight assigned to the vertices, the
* weight of the cover equals the cover's cardinality.
*/
private double upperBoundOnVertexCoverWeight;
/** Indicates whether we are solving a weighted or unweighted version of the problem **/
private boolean weighted;
private Map vertexWeightMap = null;
/////////////
/**
* Constructs a new GreedyVCImpl instance
*
* @param graph input graph
*/
public RecursiveExactVCImpl(Graph graph)
{
this.graph = GraphTests.requireUndirected(graph);
this.vertexWeightMap = graph
.vertexSet().stream().collect(Collectors.toMap(Function.identity(), vertex -> 1.0));
weighted = false;
}
/**
* Constructs a new GreedyVCImpl instance
*
* @param graph input graph
* @param vertexWeightMap mapping of vertex weights
*/
public RecursiveExactVCImpl(Graph graph, Map vertexWeightMap)
{
this.graph = GraphTests.requireUndirected(graph);
this.vertexWeightMap = Objects.requireNonNull(vertexWeightMap);
weighted = true;
}
@Override
public VertexCoverAlgorithm.VertexCover getVertexCover()
{
// Initialize
this.graph = GraphTests.requireUndirected(graph);
memo = new HashMap<>();
vertices = new ArrayList<>(graph.vertexSet());
neighborCache = new NeighborCache<>(graph);
vertexIDDictionary = new HashMap<>();
n = vertices.size();
// Sort vertices based on their weight/degree ratio in ascending order
// TODO JK: Are there better orderings?
vertices.sort(Comparator.comparingDouble(v -> vertexWeightMap.get(v) / graph.degreeOf(v)));
for (int i = 0; i < vertices.size(); i++)
vertexIDDictionary.put(vertices.get(i), i);
// Calculate a bound on the maximum depth using heuristics and mathematical bounding
// procedures.
// TODO JK: Is there a lower bounding procedure which allows us to prematurely terminate the
// search once a solution is found which is equal to the lower bound? Preferably a bounding
// procedure which gets better throughout the search.
upperBoundOnVertexCoverWeight = this.calculateUpperBound();
// Invoke recursive algorithm
BitSetCover vertexCover = this.calculateCoverRecursively(0, new BitSet(n), 0);
// Build solution
Set verticesInCover = new LinkedHashSet<>();
for (int i = vertexCover.bitSetCover.nextSetBit(0); i >= 0 && i < n;
i = vertexCover.bitSetCover.nextSetBit(i + 1))
verticesInCover.add(vertices.get(i));
return new VertexCoverAlgorithm.VertexCoverImpl<>(verticesInCover, vertexCover.weight);
}
private BitSetCover calculateCoverRecursively(
int indexNextCandidate, BitSet visited, double accumulatedWeight)
{
// Check memoization table
if (memo.containsKey(visited)) {
return memo.get(visited).copy(); // Cache hit
}
// Find the next unvisited vertex WITH neighbors (if a vertex has no neighbors, then we
// don't need to select it
// because it doesn't cover any edges)
int indexNextVertex = -1;
Set neighbors = Collections.emptySet();
for (int index = visited.nextClearBit(indexNextCandidate); index >= 0 && index < n;
index = visited.nextClearBit(index + 1))
{
neighbors = new LinkedHashSet<>(neighborCache.neighborsOf(vertices.get(index)));
for (Iterator it = neighbors.iterator(); it.hasNext();) // Exclude all visited
// vertices
if (visited.get(vertexIDDictionary.get(it.next())))
it.remove();
if (!neighbors.isEmpty()) {
indexNextVertex = index;
break;
}
}
// Base case 1: all vertices have been visited
if (indexNextVertex == -1) { // We've visited all vertices, return the base case
BitSetCover vertexCover = new BitSetCover(n, 0);
if (accumulatedWeight <= upperBoundOnVertexCoverWeight) { // Found new a solution that
// matches our bound. Tighten
// the bound.
upperBoundOnVertexCoverWeight = accumulatedWeight - 1;
}
return vertexCover;
// Base case 2 (pruning): this vertex cover can never be better than the best cover we
// already have. Return a cover with a large weight, such that the other branch will be
// preferred over this branch.
} else if (accumulatedWeight >= upperBoundOnVertexCoverWeight) {
return new BitSetCover(n, n);
}
// Recursion
// TODO JK: Can we use a lower bound or estimation which of these 2 branches produces a
// better solution? If one of them is more likely to produce a better solution,
// then that branch should be explored first! Futhermore, if the lower bound+accumulated
// cost > upperBoundOnVertexCoverWeight, then we may prune.
// Create 2 branches (N(v) denotes the set of neighbors of v. G_{v} indicates the graph
// obtained by removing vertex v and all vertices incident to it.):
// Right branch (N(v) are added to the cover, and we solve for G_{N(v) \cup v }$.):
BitSet visitedRightBranch = (BitSet) visited.clone();
visitedRightBranch.set(indexNextVertex);
for (V v : neighbors)
visitedRightBranch.set(vertexIDDictionary.get(v));
double weight = this.getWeight(neighbors);
BitSetCover rightCover = calculateCoverRecursively(
indexNextVertex + 1, visitedRightBranch, accumulatedWeight + weight);
List neighborsIndices =
neighbors.stream().map(vertexIDDictionary::get).collect(Collectors.toList());
rightCover.addAllVertices(neighborsIndices, weight);
// Left branch (vertex v is added to the cover, and we solve for G_{v}):
BitSet visitedLeftBranch = (BitSet) visited.clone();
visitedLeftBranch.set(indexNextVertex);
weight = vertexWeightMap.get(vertices.get(indexNextVertex));
BitSetCover leftCover = calculateCoverRecursively(
indexNextVertex + 1, visitedLeftBranch, accumulatedWeight + weight);
leftCover.addVertex(indexNextVertex, weight); // Delayed update of the left cover
// Return the best branch
if (leftCover.weight <= rightCover.weight) {
memo.put(visited, leftCover.copy());
return leftCover;
} else {
memo.put(visited, rightCover.copy());
return rightCover;
}
}
/**
* Returns the weight of a collection of vertices. In case of the unweighted vertex cover
* problem, the return value is the cardinality of the collection. In case of the weighted
* version, the return value is the sum of the weights of the vertices
*
* @param vertices vertices
* @return the total weight of the vertices in the collection.
*/
private double getWeight(Collection vertices)
{
if (weighted) {
return vertices.stream().map(vertexWeightMap::get).reduce(0d, Double::sum);
} else {
return vertices.size();
}
}
/**
* Calculates a cheap upper bound on the optimum solution. Currently, we return the best
* solution found by either the greedy heuristic, or Clarkson's 2-approximation. Neither of
* these 2 algorithms dominates the other. //TODO JK: Are there better bounding procedures?
*/
private double calculateUpperBound()
{
return Math
.min(
new GreedyVCImpl<>(graph, vertexWeightMap).getVertexCover().getWeight(),
new ClarksonTwoApproxVCImpl<>(graph, vertexWeightMap).getVertexCover().getWeight());
}
/**
* Helper class which represents a vertex cover as a space efficient BitSet
*/
protected class BitSetCover
{
protected BitSet bitSetCover;
protected double weight;
/**
* Construct a new empty vertex cover as a BitSet.
*
* @param size initial capacity of the BitSet
* @param initialWeight the initial weight
*/
protected BitSetCover(int size, int initialWeight)
{
bitSetCover = new BitSet(size);
this.weight = initialWeight;
}
/**
* Copy constructor
*
* @param vertexCover the input vertex cover to copy
*/
protected BitSetCover(BitSetCover vertexCover)
{
this.bitSetCover = (BitSet) vertexCover.bitSetCover.clone();
this.weight = vertexCover.weight;
}
/**
* Copy a vertex cover.
*
* @return a copy of the vertex cover
*/
protected BitSetCover copy()
{
return new BitSetCover(this);
}
/**
* Add a vertex in the vertex cover.
*
* @param vertexIndex the index of the vertex
* @param weight the weight of the vertex
*/
protected void addVertex(int vertexIndex, double weight)
{
bitSetCover.set(vertexIndex);
this.weight += weight;
}
/**
* Add multiple vertices in the vertex cover.
*
* @param vertexIndices the index of the vertices
* @param totalWeight the total weight of the vertices
*/
protected void addAllVertices(List vertexIndices, double totalWeight)
{
vertexIndices.forEach(bitSetCover::set);
this.weight += totalWeight;
}
}
}
