Add Zykov and Christofides Algorithms for Chromatic Number

doc/attr.xml

@@ -1441,6 +1441,10 @@ gap> DigraphLoops(D);
       <Cite Key="Law1976"/></Item>
       <Item><C>byskov</C> - Byskov's Algorithm 
       <Cite Key="Bys2002"/></Item>
+      <Item><C>zykov</C> - Zykov's Algorithm 
+      <Cite Key="Corneil1973"/></Item>
+      <Item><C>christofides</C> - Christofides's Algorithm 
+      <Cite Key="Wang1974"/></Item>
     </List>
 
     <Example><![CDATA[

doc/digraphs.bib

@@ -183,3 +183,35 @@ @article{Bys2002
     Journal = {BRICS Report Series},
     Doi = {10.7146/brics.v9i45.21760}
 }
+
+@article{Corneil1973,
+author = {Corneil, D. G. and Graham, B.},
+title = {An Algorithm for Determining the Chromatic Number of a Graph},
+journal = {SIAM Journal on Computing},
+volume = {2},
+number = {4},
+pages = {311-318},
+year = {1973},
+doi = {10.1137/0202026},
+URL = {https://doi.org/10.1137/0202026},
+eprint = {https://doi.org/10.1137/0202026}
+}
+
+@article{Wang1974,
+author= {Wang, Chung C.},
+title = {An Algorithm for the Chromatic Number of a Graph},
+year = {1974},
+issue_date = {July 1974},
+publisher = {Association for Computing Machinery},
+address = {New York, NY, USA},
+volume = {21},
+number = {3},
+issn = {0004-5411},
+url = {https://doi.org/10.1145/321832.321837},
+doi = {10.1145/321832.321837},
+abstract = {Christofides' algorithm for finding the chromatic number of a graph is improved both in speed and memory space by using a depth-first search rule to search for a shortest path in a reduced subgraph tree.},
+journal = {J. ACM},
+month = jul,
+pages = {385–391},
+numpages = {7}
+} 

gap/attr.gi

@@ -369,6 +369,192 @@ function(D)
 end
 );
 
+BindGlobal("DIGRAPHS_ChromaticNumberZykov",
+function(D)
+  local nr, ZykovReduce, chrom;
+  nr := DigraphNrVertices(D);
+  # Recursive function call
+  ZykovReduce := function(D)
+    local nr, D_contract, vertices, v, x, y, i, j, adjacent;
+    nr := DigraphNrVertices(D);
+    # Update upper bound if possible.
+    chrom := Minimum(nr, chrom);
+    # Leaf nodes are either complete graphs or cliques that have size equal to
+    # the current upper bound. The chromatic number is then the smallest clique
+    # found.
+    # Cliques finder arguments:
+    # digraph = D - The graph
+    # hook = fail - hook is not required
+    # user_param = [] - user_param is a list as hook is fail
+    # limit = 1 - We only need one clique
+    # include = exclude = [] - We check all vertices
+    # max = false - This clique need not be maximal
+    # size = chrom - We want a clique the size of our upper bound
+    # reps = true - As we only care about the existence of the clique,
+    # we can instead search for representatives which is more efficient.
+    if not IsCompleteDigraph(D) and IsEmpty(CliquesFinder(D, fail, [], 1, [],
+                                                          [], false, chrom,
+                                                          true)) then
+      # Sort vertices by degree, so that higher degree vertices are picked first
+      # Picking higher degree vertices will make it more likely a clique will
+      # form in one of the modified graphs, which will terminate the recursion.
+      vertices := DigraphWelshPowellOrder(D);
+      # Get the adjacency function
+      adjacent := DigraphAdjacencyFunction(D);
+      # Choose two non-adjacent vertices x, y
+      for i in [1 .. nr] do
+        x := vertices[i];
+        # Search for the first vertex not adjacent to all others
+        # This is guaranteed to exist as D is not the complete graph.
+        if OutDegreeOfVertex(D, x) < nr - 1 then
+          # Now search for a non-adjacent vertex, prioritising higher degree ones
+          for j in [i + 1 .. nr] do
+            y := vertices[j];
+            if not adjacent(x, y) then
+              break;
+            fi;
+          od;
+          break;
+        fi;
+      od;
+      Assert(1, x <> y, "x and y must be different");
+      # Colour the vertex contraction.
+      # A contraction of a graph effectively merges two non adjacent vertices
+      # into a single new vertex with the edges merged.
+      # We merge y into x, keeping x.
+
+      # TODO Use DigraphContractEdge once it is implemented.
+      D_contract := DigraphMutableCopy(D);
+      for v in vertices do
+         # Iterate over all vertices that are not x or y
+         if v = x or v = y then
+           continue;
+         fi;
+         # Add any edge that involves y, but not already x to avoid duplication.
+         if adjacent(v, y) and not adjacent(v, x) then
+            DigraphAddEdge(D_contract, x, v);
+            DigraphAddEdge(D_contract, v, x);
+         fi;
+      od;
+      DigraphRemoveVertex(D_contract, y);
+      ZykovReduce(D_contract);
+      # Colour the edge addition
+      # This just adds symmetric edges between x and y;
+      DigraphAddEdge(D, [x, y]);
+      DigraphAddEdge(D, [y, x]);
+      ZykovReduce(D);
+      # Undo changes to the graph
+      DigraphRemoveEdge(D, [x, y]);
+      DigraphRemoveEdge(D, [y, x]);
+    fi;
+  end;
+  # Algorithm requires an undirected graph without multiple edges.
+  D := DigraphMutableCopy(D);
+  D := DigraphRemoveAllMultipleEdges(D);
+  D := DigraphSymmetricClosure(D);
+  # Use greedy colouring as an upper bound
+  chrom := RankOfTransformation(DigraphGreedyColouring(D), nr);
+  ZykovReduce(D);
+  return chrom;
+end
+);
+
+BindGlobal("DIGRAPHS_ChromaticNumberChristofides",
+function(D)
+  local nr, I, n, T, b, unprocessed, i, v_without_t, j, u, min_occurrences,
+  cur_occurrences, chrom, colouring, stack, vertices;
+
+  nr := DigraphNrVertices(D);
+  vertices := List(DigraphVertices(D));
+  # Initialise the required variables.
+  # Calculate all maximal independent sets of D.
+  I := DigraphMaximalIndependentSets(D);
+  # Convert each MIS into a BList
+  I := List(I, i -> BlistList(vertices, i));
+  # Upper bound for chromatic number.
+  chrom := nr;
+  # Set of vertices of D not in the current subgraph at level n.
+  T := ListWithIdenticalEntries(nr, false);
+  # Current search level of the subgraph tree.
+  n := 0;
+  # The maximal independent sets of V \ T at level n.
+  b := [ListWithIdenticalEntries(nr, false)];
+  # Number of unprocessed MIS's of V \ T from level 1 to n
+  unprocessed := ListWithIdenticalEntries(nr, 0);
+  # Would be jth colour class of the chromatic colouring of G.
+  colouring := List([1 .. nr], i -> BlistList(vertices, [i]));
+  # Stores current unprocessed MIS's of V \ T at level 1 to level n
+  stack := [];
+  # Now perform the search.
+  repeat
+    # Step 2
+    if n < chrom then
+      # Step 3
+      # If V = T then we've reached a null subgraph
+      if SizeBlist(T) = nr then
+        chrom := n;
+        SubtractBlist(T, b[n + 1]);
+        for i in [1 .. chrom] do
+          colouring[i] := b[i];
+          # TODO set colouring attribute
+        od;
+      else
+        # Step 4
+        # Compute the maximal independent sets of V \ T
+        v_without_t := DIGRAPHS_MaximalIndependentSetsSubtractedSet(I, T,
+                                                                    infinity);
+        # Step 5
+        # Pick u in V \ T such that u is in the fewest maximal independent sets.
+        u := -1;
+        min_occurrences := infinity;
+        # Flip T to get V \ T
+        FlipBlist(T);
+        # Convert to list to iterate over the vertices.
+        for i in ListBlist(vertices, T) do
+          # Count how many times this vertex appears in a MIS
+          cur_occurrences := Number(v_without_t, j -> j[i]);
+          if cur_occurrences < min_occurrences then
+            min_occurrences := cur_occurrences;
+            u := i;
+          fi;
+        od;
+        # Revert changes to T
+        FlipBlist(T);
+        Assert(1, u <> -1, "Vertex must be picked");
+        # Remove maximal independent sets not containing u.
+        v_without_t := Filtered(v_without_t, x -> x[u]);
+        # Add these MISs to the stack
+        Append(stack, v_without_t);
+        # Search has moved one level deeper
+        n := n + 1;
+        unprocessed[n] := Length(v_without_t);
+      fi;
+    else
+      # if n >= g then T = T \ b[n]
+      # This exceeds the current best bound, so stop search.
+      SubtractBlist(T, b[n + 1]);
+    fi;
+    # Step 6
+    while n <> 0 do
+      # step 7
+      if unprocessed[n] = 0 then
+        n := n - 1;
+        SubtractBlist(T, b[n + 1]);
+      else
+        # Step 8
+        # take an element from the top of the stack
+        i := Remove(stack);
+        unprocessed[n] := unprocessed[n] - 1;
+        b[n + 1] := i;
+        UniteBlist(T, i);
+        break;
+      fi;
+    od;
+  until n = 0;
+  return chrom;
+end
+);
+
 InstallMethod(ChromaticNumber, "for a digraph by out-neighbours",
 [IsDigraphByOutNeighboursRep],
 function(D)
@@ -392,6 +578,10 @@ function(D)
     return DIGRAPHS_ChromaticNumberLawler(D);
   elif ValueOption("byskov") <> fail then
     return DIGRAPHS_ChromaticNumberByskov(D);
+  elif ValueOption("zykov") <> fail then
+    return DIGRAPHS_ChromaticNumberZykov(D);
+  elif ValueOption("christofides") <> fail then
+    return DIGRAPHS_ChromaticNumberChristofides(D);
   fi;
 
   # The chromatic number of <D> is at least 3 and at most nr

tst/standard/attr.tst

@@ -1662,6 +1662,118 @@ Error, the argument <D> must be a digraph with no loops,
 gap> DIGRAPHS_UnderThreeColourable(EmptyDigraph(0));
 0
 
+#  Test ChromaticNumber Zykov
+gap> ChromaticNumber(NullDigraph(10) : zykov);
+1
+gap> ChromaticNumber(CompleteDigraph(10) : zykov);
+10
+gap> ChromaticNumber(CompleteBipartiteDigraph(5, 5) : zykov);
+2
+gap> ChromaticNumber(DigraphRemoveEdge(CompleteDigraph(10), [1, 2]) : zykov);
+10
+gap> ChromaticNumber(Digraph([[4, 8], [6, 10], [9], [2, 3, 9], [],
+> [3], [4], [6], [], [5, 7]]) : zykov);
+3
+gap> ChromaticNumber(DigraphDisjointUnion(CompleteDigraph(1),
+> Digraph([[2], [4], [1, 2], [3]])) : zykov);
+3
+gap> ChromaticNumber(DigraphDisjointUnion(CompleteDigraph(1),
+> Digraph([[2], [4], [1, 2], [3], [1, 2, 3]])) : zykov);
+4
+gap> gr := Digraph([[2, 3, 4], [3], [], []]);
+<immutable digraph with 4 vertices, 4 edges>
+gap> ChromaticNumber(gr : zykov);
+3
+gap> ChromaticNumber(EmptyDigraph(0) : zykov);
+0
+gap> gr := CompleteDigraph(4);;
+gap> gr := DigraphAddVertex(gr);;
+gap> ChromaticNumber(gr : zykov);
+4
+gap> gr := Digraph([[2, 4, 7, 3], [3, 5, 8, 1], [1, 6, 9, 2],
+> [5, 7, 1, 6], [6, 8, 2, 4], [4, 9, 3, 5], [8, 1, 4, 9], [9, 2, 5, 7],
+> [7, 3, 6, 8]]);;
+gap> ChromaticNumber(gr : zykov);
+3
+gap> gr := DigraphSymmetricClosure(ChainDigraph(5));
+<immutable symmetric digraph with 5 vertices, 8 edges>
+gap> ChromaticNumber(gr : zykov);
+2
+gap> gr := DigraphFromGraph6String("KmKk~K??G@_@");
+<immutable symmetric digraph with 12 vertices, 42 edges>
+gap> ChromaticNumber(gr : zykov);
+4
+gap> gr := CycleDigraph(7);
+<immutable cycle digraph with 7 vertices>
+gap> ChromaticNumber(gr : zykov);
+3
+gap> ChromaticNumber(gr : zykov);
+3
+gap> ChromaticNumber(gr : zykov);
+3
+gap> a := DigraphRemoveEdges(CompleteDigraph(50), [[1, 2], [2, 1]]);;
+gap> b := DigraphAddVertex(a);;
+gap> ChromaticNumber(a : zykov);
+49
+gap> ChromaticNumber(b : zykov);
+49
+
+#  Test ChromaticNumber Christofides
+gap> ChromaticNumber(NullDigraph(10) : christofides);
+1
+gap> ChromaticNumber(CompleteDigraph(10) : christofides);
+10
+gap> ChromaticNumber(CompleteBipartiteDigraph(5, 5) : christofides);
+2
+gap> ChromaticNumber(DigraphRemoveEdge(CompleteDigraph(10), [1, 2]) : christofides);
+10
+gap> ChromaticNumber(Digraph([[4, 8], [6, 10], [9], [2, 3, 9], [],
+> [3], [4], [6], [], [5, 7]]) : christofides);
+3
+gap> ChromaticNumber(DigraphDisjointUnion(CompleteDigraph(1),
+> Digraph([[2], [4], [1, 2], [3]])) : christofides);
+3
+gap> ChromaticNumber(DigraphDisjointUnion(CompleteDigraph(1),
+> Digraph([[2], [4], [1, 2], [3], [1, 2, 3]])) : christofides);
+4
+gap> gr := Digraph([[2, 3, 4], [3], [], []]);
+<immutable digraph with 4 vertices, 4 edges>
+gap> ChromaticNumber(gr : christofides);
+3
+gap> ChromaticNumber(EmptyDigraph(0) : christofides);
+0
+gap> gr := CompleteDigraph(4);;
+gap> gr := DigraphAddVertex(gr);;
+gap> ChromaticNumber(gr : christofides);
+4
+gap> gr := Digraph([[2, 4, 7, 3], [3, 5, 8, 1], [1, 6, 9, 2],
+> [5, 7, 1, 6], [6, 8, 2, 4], [4, 9, 3, 5], [8, 1, 4, 9], [9, 2, 5, 7],
+> [7, 3, 6, 8]]);;
+gap> ChromaticNumber(gr : christofides);
+3
+gap> gr := DigraphSymmetricClosure(ChainDigraph(5));
+<immutable symmetric digraph with 5 vertices, 8 edges>
+gap> ChromaticNumber(gr : christofides);
+2
+gap> gr := DigraphFromGraph6String("KmKk~K??G@_@");
+<immutable symmetric digraph with 12 vertices, 42 edges>
+gap> ChromaticNumber(gr : christofides);
+4
+gap> gr := CycleDigraph(7);
+<immutable cycle digraph with 7 vertices>
+gap> ChromaticNumber(gr : christofides);
+3
+gap> ChromaticNumber(gr : christofides);
+3
+gap> ChromaticNumber(gr : christofides);
+3
+gap> a := DigraphRemoveEdges(CompleteDigraph(50), [[1, 2], [2, 1]]);;
+gap> b := DigraphAddVertex(a);;
+gap> ChromaticNumber(a : christofides);
+49
+gap> ChromaticNumber(b : christofides);
+49
+
 #  DegreeMatrix
 gap> gr := Digraph([[2, 3, 4], [2, 5], [1, 5, 4], [1], [1, 1, 2, 4]]);;
 gap> DegreeMatrix(gr);