Add an acyclic coloring

[amontoison]

modified by @gdalle

• Implement acyclic_coloring with the efficient tree-based algorithm (adds DataStructures.jl to the deps).
• Implement very naive decompression by substitution for symmetric matrices, can be optimized much more even without the TreeSet. It works by building a matrix T such that T * upper_nonzeros(A) == vec(B) and then inverting the system.
• Add tests on small instances from the papers as well as large random instances (the equality test is only approximate).

[gdalle]

Thanks for contributing this! However, before adding it, we may want to think about decompression and how to implement it, see #35 and #37

[amontoison]

@gdalle
I will need your help to find the correct size to initialize first_visit_to_tree.
Can we do better than first_visit_to_tree = fill((0, 0), nnz(g))?
I don't know if we can have cases with #trees == #edges.

I have a functional function acyclic_coloring but it doesn't return a correct coloring yet.

using SparseMatrixColorings

A = [
   1  2  0  0  0  0  3  0  0  0
   2  4  5  0  6  0  0  0  0  0
   0  5  7  8  0  9  0  0  0  0
   0  0  8  10 0  0  0  0  0  11
   0  6  0  0  12 13 0  14 0  0
   0  0  9  0  13 15 0  0  16 0
   3  0  0  0  0  0  17 18 0  0
   0  0  0  0  14 0  18 19 20 0
   0  0  0  0  0  16 0  20 21 22
   0  0  0  11 0  0  0  0  22 23
]

g = SparseMatrixColorings.Graph(A)

colors = SparseMatrixColorings.acyclic_coloring(g, SparseMatrixColorings.NaturalOrder())

@test colors == [1, 2, 1, 2, 1, 3, 2, 3, 2, 1]

[amontoison]

I checked the code in JuMP, and they use the number of edges:
https://github.com/jump-dev/MathOptInterface.jl/blob/master/src/Nonlinear/ReverseAD/Coloring/Coloring.jl#L184-L187.
They also initialize disjoint_sets with the number of edges instead of dynamically using push! like I do.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 99.41%. Comparing base (4de73c3) to head (d080331).

Additional details and impacted files

@@            Coverage Diff             @@
##             main      #36      +/-   ##
==========================================
+ Coverage   99.23%   99.41%   +0.18%     
==========================================
  Files          11       11              
  Lines         394      516     +122     
==========================================
+ Hits          391      513     +122     
  Misses          3        3              

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[gdalle]

I merged #38 even though I wanted to add a few more things so that you can rebase this PR off of it

[gdalle]

I merged the new ColoringProblem interface from #45 into this PR

[gdalle]

I merged a few more PRs into this branch, including one that cleans up the examples from the paper figures. Interestingly, Figures 6.1 from "What" and Figure 4 from "Efficient" have the same matrix but different colorings, so it's expected that you wouldn't find both

[amontoison]

I merged a few more PRs into this branch, including one that cleans up the examples from the paper figures. Interestingly, Figures 6.1 from "What" and Figure 4 from "Efficient" have the same matrix but different colorings, so it's expected that you wouldn't find both

I remarked that but the coloring of Figure 6.1 is just an example of an acyclic coloring, it's not related to any code so I don't obtain it with my implementation.
However it's a nice example for testing decompression.