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Add Zykov and Christofides Algorithms for Chromatic Number #491

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Add Zykov and Christofides Algorithms for Chromatic Number #491

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101c55d
Add Zykov and Christofides Algorithms for Chromatic Number
EwanGilligan Jun 2, 2021
0a8144a
Merge branch 'master' into egilligan-bab-chromatic-number
james-d-mitchell Nov 30, 2021
0af7fc2
Make requested changes
EwanGilligan Jan 13, 2022
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4 changes: 4 additions & 0 deletions doc/attr.xml
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Expand Up @@ -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[
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32 changes: 32 additions & 0 deletions doc/digraphs.bib
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Expand Up @@ -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}
}
179 changes: 179 additions & 0 deletions gap/attr.gi
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Expand Up @@ -369,6 +369,181 @@ 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, adjacent, vertices, v, x, y, x_i, y_i, found, deg;
nr := DigraphNrVertices(D);
# Update upper bound if possible.
chrom := Minimum(nr, chrom);
# Leaf nodes are either complete graphs or q-cliques. The chromatic number
# is then the smallest q-clique found.
if not IsCompleteDigraph(D) and IsEmpty(CliquesFinder(D, fail, [], 1, [],
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Could you possibly add a comment to this line about what it is that the CliquesFinder function is looking for here?

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I'll add some additional explanation for the CliqueFinder call.

[], false, chrom,
true)) then
# Get adjacency function
adjacent := DigraphAdjacencyFunction(D);
# Sort vertices by degree, so that higher degree vertices are picked first
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This is already implemented in DigraphWelshPowellOrder maybe you could use that instead of rolling your own?

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Thank you for letting me know about DigraphWelshPowellOrder, I'll make the change to use that instead!

vertices := [1 .. nr];
deg := ShallowCopy(OutDegrees(D));
SortParallel(deg, vertices, {x, y} -> x > y);
# Choose two non-adjacent vertices x, y
# This is just done by ascending ordering.
found := false;
for x_i in [1 .. nr] do
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This seems like a lot of work to find two non-adjacent vertices, couldn't you just take the last node u in the list vertices (which has lowest degree) and take the any value not in OutNeighbours(D, u);? This has to exist because u is of minimum degree and D is not a complete graph.

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The reasoning behind the approach is that the recursion will terminate either when it finds a clique the size of the current upper bound or we reach the complete graph. By prioritising high degree vertices, it is more likely we will reach a clique in fewer steps. I'd be happy to run a few experiments though to test if you think it is worth checking.

x := vertices[x_i];
for y_i in [x_i + 1 .. nr] do
y := vertices[y_i];
if not adjacent(x, y) then
found := true;
break;
fi;
od;
if found then
break;
fi;
od;
Assert(1, x <> y, "x and y must be different");
Assert(1, found, "No adjacent vertices");
# 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.
D_contract := DigraphMutableCopy(D);
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Can't you use DigraphContractEdge to do this step too?

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Looking through the manual I can't seem to find any reference to DigraphContractEdge, but I believe it would be possible to replace the step with DigraphQuotient by taking the partition the two adjacent vertices in one part, and then the remaining vertices are singleton parts. I need to look into the implementation of DigraphQuotient, as it seems more general that what is needed here and so could be slower.

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I see now there is an open pull request for DigraphContractEdge, so I'll use that once the required changes are made.

for v in vertices do
# Iterate over all vertices that
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.
D := DigraphSymmetricClosure(DigraphMutableCopy(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_occurences,
cur_occurences, 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_occurences := infinity;
for i in vertices do
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This seems to have unnecessarily high complexity. Why not loop over everything in v_without_t one time to find a list whose i-th entry is the number of times it occurs in the an entry of v_without_t?

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That is a more sensible approach, I'll make that change now

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I experimented with both this approach and the current approach, and it seemed to not have a noticeable difference. I've instead simplified the current approach, so that it should be clearer and neater.

# Skip elements of T.
if T[i] then
continue;
fi;
cur_occurences := 0;
for j in v_without_t do
if j[i] then
cur_occurences := cur_occurences + 1;
fi;
od;
if cur_occurences < min_occurences then
min_occurences := cur_occurences;
u := i;
fi;
od;
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)
Expand All @@ -392,6 +567,10 @@ 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
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112 changes: 112 additions & 0 deletions tst/standard/attr.tst
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Expand Up @@ -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);
Expand Down