// Removes a separator from the graph, and recursively computes a tree decomposition. The separator should be safe.
// If check3isolated is true, then this also checks the property that given a separator of size 3, it should not split off any isolated vertices (or else it is not safe).
// If any connected component has more than maximumSize vertices, do not proceed (balanced separator check)
public Graph RemoveSeparator(List <Vertex> separator, bool check3isolated, int maximumSize)
{
// Compute the graph after removing separator
List <Vertex> tempVertices = vertices.Keys.ToList();
foreach (Vertex v in separator)
{
tempVertices.Remove(v);
}
// Get the connected components
Graph componentsGraph = new Graph(tempVertices);
List <List <Vertex> > components = componentsGraph.getComponents();
// If applicable, check that it is safe
if (check3isolated && components.Any((c) => c.Count == 1))
{
return(null);
}
// Check balance
if (components.Any((c) => c.Count > maximumSize))
{
return(null);
}
// Prepare the new tree decomposition: the root bag contains the separator
Graph result = new Graph();
Program.BagsList.Add(new List <Vertex>(separator));
Vertex root = result.AddVertex(Program.BagsList.Count - 1);
// Decompose each component
foreach (List <Vertex> component in components)
{
// Create a new graph with the component in it
Graph componentGraph = new Graph(component);
// Add the separator
foreach (Vertex v in separator)
{
componentGraph.AddVertex(v);
}
foreach (Vertex u in separator)
{
// Make the separator a clique
foreach (Vertex v in separator)
{
if (u.Label <= v.Label) // Not strictly necessary, but might save some time
{
continue;
}
componentGraph.AddEdge(u, v);
}
// And add edges from the separator to the rest
foreach (Vertex v in u.Adj)
{
componentGraph.AddEdge(u, v);
}
}
// Compute the TD of the component
Graph TDComponent = componentGraph.Decompose();
result.AddGraph(TDComponent);
// Find an appropriate bag in the decomposition to connect our root bag to
foreach (Vertex v in TDComponent.vertices.Keys)
{
if (separator.All((u) => { return(Program.BagsList[v.Label].Contains(u)); }))
{
result.AddEdge(root, v);
break;
}
}
}
return(result);
}
// Attempt to branch on a separator
public Graph BranchOnSeparator()
{
// Attempt to branch on a separator
// Theorem: a graph of treewidth k has a 0.5(n-k) separator of size at most k+1
int bestSeparatorSize = vertices.Count;
List <Vertex> bestSeparator = new List <Vertex>(vertices.Keys);
List <List <Vertex> > separators = new List <List <Vertex> >();
foreach (Vertex v in vertices.Keys) // Initialization: all vertex subsets of size 1
{
separators.Add(new List <Vertex>(new Vertex[] { v }));
}
int lower = Program.BagsList.Count > 0 ? Program.BagsList.Max((b) => b.Count) : 0;
// We enumerate over all vertex subsets of increasing size, and check whether they form a balanced separator (n choose k options)
// If so, we branch on removing this separator
// We consider all subsets of size k-1, and extend it to a subset of size k by adding an articulation point
// This reduces the number of points we have to consider (in the last step) greatly
// One caveat: if a component was off by 1 from the upper, limit, we have to consider all possibilities for this component
for (int i = 2; i < bestSeparatorSize - 1; i++)
{
List <List <Vertex> > newSeparators = new List <List <Vertex> >();
foreach (List <Vertex> separator in separators) // Enumerate over separators of size i-1
{
foreach (Vertex v in vertices.Keys)
{
if (v.Label <= separator[separator.Count - 1].Label)
{
continue; // Optimization: only consider separators where the vertices are in increasing order
}
List <Vertex> newSeparator = new List <Vertex>(separator);
newSeparator.Add(v);
newSeparators.Add(newSeparator);
// Copy the graph, remove potential separator
List <Vertex> tempVertices = vertices.Keys.ToList();
foreach (Vertex u in newSeparator)
{
tempVertices.Remove(u);
}
// Consider the possible articulation points that can extend this set to a separator
// OR vertices of components that are just marginally too large
IEnumerable <Vertex> aps = new Graph(tempVertices).ArticulationPointsOrComponentSize((vertices.Count - newSeparator.Count + 2) / 2);
// Try each one as a possible separator
foreach (Vertex ap in aps)
{
if (ap.Label <= separator[separator.Count - 1].Label)
{
continue; // This one will be generated some other way as well
}
Program.SeparatorTried++;
newSeparator.Add(ap);
int oldBagCount = Program.BagsList.Count;
// There is (guaranteed) a balanced separator (biggest component at most (n-k+1)/2 vertices)
// We have to round this number up because of the adj-trick
Graph candidateDecomposition = RemoveSeparator(newSeparator, (vertices.Count - newSeparator.Count + 1) / 2);
if (candidateDecomposition != null)
{
int candidateWidth = Program.BagsList.Max((b) => b.Count);
if (candidateWidth == newSeparator.Count || candidateWidth <= lower)
{
return(RemoveSeparator(newSeparator));
}
if (candidateWidth < bestSeparatorSize)
{
bestSeparatorSize = candidateWidth;
bestSeparator = new List <Vertex>(newSeparator);
}
}
Program.BagsList.RemoveRange(oldBagCount, Program.BagsList.Count - oldBagCount);
newSeparator.RemoveAt(newSeparator.Count - 1);
if (newSeparator.Count + 1 == bestSeparatorSize)
{
return(RemoveSeparator(bestSeparator));
}
#if LIMIT_SEPARATOR
if (Program.TotalExpanded > 30000000 || Program.SeparatorTried > 12000000)
{
return(null); // Apparently finding a separator in this graph is really hard, don't know how to solve, just fall back to DFS/DP and hope for the best
}
#endif
}
}
}
separators = newSeparators;
}
return(RemoveSeparator(bestSeparator));
}