added fractional_chromatic_number, fractional_clique_number (#13)

* added fractional_chromatic_number, fractional_clique_number

* fractional_coloring: use HiGHS rather than GLPK

* fractional_coloring: various fixes

* Add quotes

* format

---------

Co-authored-by: Guillaume Dalle <22795598+gdalle@users.noreply.github.com>

Project.toml

@@ -30,6 +30,7 @@ Aqua = "4c88cf16-eb10-579e-8560-4a9242c79595"
 Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4"
 Graphs = "86223c79-3864-5bf0-83f7-82e725a168b6"
 HiGHS = "87dc4568-4c63-4d18-b0c0-bb2238e4078b"
+IterTools = "c8e1da08-722c-5040-9ed9-7db0dc04731e"
 JET = "c3a54625-cd67-489e-a8e7-0a5a0ff4e31b"
 JuMP = "4076af6c-e467-56ae-b986-b466b2749572"
 JuliaFormatter = "98e50ef6-434e-11e9-1051-2b60c6c9e899"
@@ -38,4 +39,4 @@ SparseArrays = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"
 Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
 
 [targets]
-test = ["Aqua", "Documenter", "Graphs", "HiGHS", "JET", "JuMP", "JuliaFormatter", "LinearAlgebra", "Test", "SparseArrays"]
+test = ["Aqua", "Documenter", "Graphs", "HiGHS", "IterTools", "JET", "JuMP", "JuliaFormatter", "LinearAlgebra", "Test", "SparseArrays"]

docs/src/algorithms.md

@@ -71,6 +71,13 @@ graph_matching
 GraphsOptim.graph_matching_step_size
 ```
 
+## Coloring
+
+```@docs
+fractional_chromatic_number
+fractional_clique_number
+```
+
 ## Utils
 
 ```@docs
@@ -80,4 +87,5 @@ GraphsOptim.is_stochastic
 GraphsOptim.is_doubly_stochastic
 GraphsOptim.is_permutation_matrix
 GraphsOptim.flat_doubly_stochastic
+GraphsOptim.indvec
 ```

src/GraphsOptim.jl

@@ -7,11 +7,13 @@ module GraphsOptim
 
 using Graphs: AbstractGraph, is_directed
 using Graphs: vertices, edges, nv, ne, src, dst, inneighbors, outneighbors
+using Graphs: complement, maximal_cliques
 using FillArrays: Zeros, Ones, Fill
 using HiGHS: HiGHS
 using JuMP: Model, AffExpr
 using JuMP: objective_function, add_to_expression!
 using JuMP: set_silent, optimize!, termination_status, value
+using JuMP: set_optimizer, objective_value
 using JuMP: @variable, @constraint, @objective
 using LinearAlgebra: norm, tr, dot
 using MathOptInterface: OPTIMAL
@@ -21,12 +23,14 @@ using OptimalTransport: sinkhorn
 export min_cost_flow
 export min_cost_assignment
 export FAQ, GOAT, graph_matching
+export fractional_chromatic_number, fractional_clique_number
 export shortest_path
 
 include("utils.jl")
 include("flow.jl")
 include("assignment.jl")
 include("graph_matching.jl")
+include("fractional_coloring.jl")
 include("shortest_path.jl")
 
 end

src/fractional_coloring.jl

@@ -0,0 +1,66 @@
+"""
+    fractional_chromatic_number(g; optimizer)
+
+Compute the fractional chromatic number of a graph.  Gives the same result as
+`fractional_clique_number`, though one function may run faster than the other.
+Beware: this can run very slowly for graphs of any substantial size.
+
+# Keyword arguments
+
+- `optimizer`: JuMP-compatible solver (default is `HiGHS.Optimizer`)
+
+# References
+
+- https://mathworld.wolfram.com/FractionalChromaticNumber.html
+"""
+function fractional_chromatic_number(
+    g::AbstractGraph{T}, optimizer=HiGHS.Optimizer
+) where {T<:Integer}
+    if is_directed(g)
+        throw(ArgumentError("The graph must not be directed"))
+    end
+
+    ss = maximal_cliques(complement(g))
+    M = hcat(indvec.(ss, nv(g))...)
+
+    model = Model(optimizer)
+    set_silent(model)
+    @variable(model, x[1:length(ss)] >= 0)
+    @constraint(model, M * x .>= 1)
+    @objective(model, Min, sum(x))
+    optimize!(model)
+    return objective_value(model)
+end
+
+"""
+    fractional_clique_number(g; optimizer)
+
+Compute the fractional clique number of a graph.  Gives the same result as
+`fractional_chromatic_number`, though one function may run faster than the other.
+Beware: this can run very slowly for graphs of any substantial size.
+
+# Keyword arguments
+
+- `optimizer`: JuMP-compatible solver (default is `HiGHS.Optimizer`)
+
+# References
+
+- https://mathworld.wolfram.com/FractionalCliqueNumber.html
+"""
+function fractional_clique_number(
+    g::AbstractGraph{T}, optimizer=HiGHS.Optimizer
+) where {T<:Integer}
+    if is_directed(g)
+        throw(ArgumentError("The graph must not be directed"))
+    end
+
+    model = Model(optimizer)
+    set_silent(model)
+    @variable(model, x[1:nv(g)] >= 0)
+    for clique in maximal_cliques(complement(g))
+        @constraint(model, sum(x[clique]) <= 1)
+    end
+    @objective(model, Max, sum(x))
+    optimize!(model)
+    return objective_value(model)
+end

src/utils.jl

@@ -48,3 +48,14 @@ end
 Return the barycenter of doubly stochastic matrices `J = 𝟏 * 𝟏ᵀ / n`.
 """
 flat_doubly_stochastic(n::Integer) = ones(n) * ones(n)' / n
+
+"""
+    indvec(s, n)
+
+Return a vector of length `n` with ones at indices specified by `s`.
+"""
+function indvec(s::AbstractVector, n::Integer)
+    x = zeros(n)
+    x[s] .= 1
+    return x
+end

test/fractional_coloring.jl

@@ -0,0 +1,12 @@
+using Graphs
+using GraphsOptim
+using IterTools
+using Test
+
+function kneser_graph(n::Integer, k::Integer)
+    ss = collect(subsets(1:n, k))
+    return SimpleGraph([isdisjoint(a, b) for a in ss, b in ss])
+end
+
+@test fractional_chromatic_number(kneser_graph(8, 3)) ≈ 8 / 3
+@test fractional_clique_number(kneser_graph(8, 3)) ≈ 8 / 3

test/runtests.jl

@@ -40,6 +40,10 @@ using Test
         include("graph_matching.jl")
     end
 
+    @testset verbose = true "Fractional coloring" begin
+        include("fractional_coloring.jl")
+    end
+
     @testset verbose = true "Shortest path" begin
         include("shortest_path.jl")
     end