// Counts the number of independent sets in a graph, such that:
// - The vertices in P are legal choices for our IS, initially set to all vertices in the graph
// - None of the vertices in X are used, initially empty
// The performDFS boolean is used to check if we should perform a check for connectedness on this level of the recursion
private static long Compute(Graph graph, BitSet p, BitSet x, bool performDfs)
{
// Base case, when P and X are both empty we cannot expand the IS
if (p.IsEmpty && x.IsEmpty)
return 1;
// Base case, if a vertex w in X has no neighbor in P, then it means that this IS will never get maximal
// since we could always include w. Thus, the IS will not be valid and we return 0.
foreach (int w in x)
if ((graph.OpenNeighborhood(w) * p).IsEmpty)
return 0;
long count = 0;
// If a DFS is needed we check if the graph induced by (P + X) is still connected.
// If the graph is disconnected, in components c1,...,cn then we can simply count the IS of all these components
// after which we simply multiply these numbers.
if (performDfs)
{
if (!DepthFirstSearch.Connected(graph, p + x))
{
count = 1;
foreach (BitSet component in DepthFirstSearch.ConnectedComponents(graph, p + x))
count *= Compute(graph, component * p, component * x, false);
return count;
}
}
// Select a pivot in P to branch on
// In this case we pick the vertex with the largest degree
int maxDegree = -1; ;
int pivot = -1;
foreach (int u in p)
{
int deg = graph.Degree(u);
if (deg > maxDegree)
{
maxDegree = deg;
pivot = u;
}
}
// There should always be a pivot after the selection procedure
if (pivot == -1)
throw new Exception("Pivot has not been selected");
// We branch on the pivot, one branch we include the pivot in the IS.
// This might possibly disconnect the subgraph G(P + X), thus we set the performDFS boolean to true.
count = Compute(graph, p - graph.ClosedNeighborhood(pivot), x - graph.OpenNeighborhood(pivot), true);
// One branch we exclude the pivot of the IS. This will not cause the graph to get possibly disconnected
count += Compute(graph, p - pivot, x + pivot, false);
return count;
}
public IEnumerable<IReductionRuleCommand> Find(Graph graph)
{
foreach (int v in graph.Vertices)
{
foreach (int u in graph.Vertices)
{
if (v == u)
{
break;
}
// TODO: Count is not O(1)
if (graph.Vertices.Count > 1 && ((graph.OpenNeighborhood(v) - u).Equals(graph.OpenNeighborhood(u) - v)))
{
graph.RemoveVertex(v);
yield return new Command(v, u);
break;
}
}
}
}
public IEnumerable<IReductionRuleCommand> Find(Graph graph)
{
foreach (int vertex in graph.Vertices)
{
// TODO: Count is not O(1)
if (graph.Degree(vertex) == 1 && graph.Vertices.Count > 1)
{
BitSet connection = graph.OpenNeighborhood(vertex);
graph.RemoveVertex(vertex);
yield return new Command(vertex, connection);
}
}
}
// Builds a neighborhood for a set X from the ground on up, without relying on what has been saved previously in O(n^2) time
public DNeighborhood(BitSet x, BitSet vector, Graph graph)
{
Vector = vector;
_occurrences = new Dictionary<int, int>();
// Loops in O(|Vector|) time over all vertices in the vector
foreach (int v in Vector)
{
// Bitset operation of O(n) time
BitSet nv = graph.OpenNeighborhood(v) * x;
_occurrences[v] = Math.Min(nv.Count, dValue);
}
}
// This constructor returns a new bipartite graph by putting all vertices in 'left' on one side, and 'right' on the other side
// There will be an edge between two vertices if there was an edge in the original graph
public BipartiteGraph(Graph graph, BitSet left, BitSet right)
: this(left, right, left.Count + right.Count)
{
Dictionary<int, int> mapping = new Dictionary<int, int>();
int i = 0;
foreach (int v in Left + Right)
mapping[v] = i++;
foreach (int v in Left)
foreach (int w in graph.OpenNeighborhood(v) * Right)
Connect(mapping[v], mapping[w]);
}
// Counts the number of independent sets in a graph, such that:
// - All vertices in R are included in the independent set, initially empty
// - Some of the vertices in P are included in the independent set, initially all vertices of the graph
// - None of the vertices in X are included in the independent set, initially empty
private static long Compute(Graph graph, BitSet r, BitSet p, BitSet x)
{
// Base case, when P and X are both empty we cannot expand the IS
if (p.IsEmpty && x.IsEmpty)
return 1;
long count = 0;
int pivot = -1;
int min = int.MaxValue;
// Procedure to find a pivot
// The idea is that in any maximal IS, either vertex u or a neighbor of u is included (else we could expand by adding u to the IS)
// We find the u with the smallest neighborhood, so that we will keep the number of branches low
foreach (int u in (p + x))
{
int size = (p * graph.OpenNeighborhood(u)).Count;
if (size < min)
{
min = size;
pivot = u;
}
}
// There should always be a pivot after the selection procedure, else P and X should both have been empty
if (pivot == -1)
throw new Exception("Pivot has not been selected");
// Foreach vertex v in the set containing the legal choices of the the closed neighborhood of the pivot,
// we include each choice in the IS and compute how many maximal IS will include v by going into recursion
foreach (int v in (p * graph.ClosedNeighborhood(pivot)))
{
count += Compute(graph, r + v, p - graph.ClosedNeighborhood(v), x - graph.OpenNeighborhood(v));
p -=v;
x += v;
}
return count;
}
// Counts the number of independent sets in a graph, such that:
// - All vertices in R are included in the independent set, initially empty
// - Some of the vertices in P are included in the independent set, initially all vertices of the graph
// - None of the vertices in X are included in the independent set, initially empty
private static int Compute(Graph graph, BitSet r, BitSet p, BitSet x)
{
// Base case, when P and X are both empty we cannot expand the IS
if (p.IsEmpty && x.IsEmpty)
return 1;
int count = 0;
BitSet copy = p;
// Foreach vertex v in P we include it in the IS and compute how many maximal IS will include v by going into recursion.
foreach (int v in copy)
{
count += Compute(graph, r + v, p - graph.ClosedNeighborhood(v), x - graph.OpenNeighborhood(v));
p -= v;
x += v;
}
return count;
}
// Given a vertex w, we can update the dNeighborhood of a set X to reflect the dNeighborhood of the set X + w in O(n) time
public DNeighborhood CopyAndUpdate(Graph graph, int w)
{
// initialize an empty dNeighborhood in O(|Vector|) time
DNeighborhood nx = new DNeighborhood(Vector);
// Copy all values in O(|Vector|) time
foreach (int v in Vector)
nx._occurrences[v] = _occurrences[v];
// Foreach vertex in N(w) * Vector they will appear one extra time in the multiset
// This operation take O(n) time because of the bitset operation
foreach (int v in graph.OpenNeighborhood(w) * Vector)
nx._occurrences[v] = Math.Min(dValue, nx._occurrences[v] + 1);
return nx;
}
