To refine our solutions obtained using heuristics and try to prove the optimality of solutions, let us formulate the Graph Coloring Problem as an ILP model. Remember it might be unable to handle large instances though. The model presented in this section and other exact algorithms are presented in *Chapter 3* of Lewis (2021).

Let us define the *Sets *considered in this approach:

*C*: Colors*N*: Nodes (or vertices)*E*: Edges

A first question already comes up “How many colors should we consider?”. In the worst-case scenario, all nodes are connected, so one should consider *C* of the same size as *N*. However, this approach could make our solutions even harder by unnecessarily increasing the number of decision variables and worsening the linear relaxation of the model. A good alternative is to use a heuristic method (such as *DSatur*) to obtain an upper bound for the number of colors.

In this problem, we have two groups of decision variables:

*x_*{*i*,*c*}: Are binary variables indicating that node*i*is colored in*c**y_*{*c*}: Are binary variables indicating that color*c*is used

We must also create constraints to ensure that:

- Every node must be colored
- If any node from one edge has a color, ensure that the color is being used
- At most 1 node of each edge can be colored with a given color
- Break symmetry (this is not required, but might help)

Finally, our objective should minimize the total number of colors used, which is the sum of *y*. Summarizing we have the following equations.

Without further ado, let us import *pyomo* for the Integer Programming model.

`import pyomo.environ as pyo`

There are two approaches for modeling a problem in *pyomo*: *Abstract *and *Concrete *models. In the first approach, the algebraic expressions of the problem are defined before some data values are supplied, whereas, in the second, the model instance is created immediately as its elements are defined. You can find more about these approaches in the library documentation or in the book by Bynum et al. (2021). Throughout this article, we will adopt the *Concrete* model formulation.

`model = pyo.ConcreteModel()`

Then, let us instantiate our *Sets*. Parsing the iterables directly from `dsatur`

attributes `N`

and `C`

is valid, so one could use them in the `initialize`

keyword arguments. Alternatively, I will pass the original *nodes *and *edges *from our input data and create a range from the *DSatur* as our initialization for colors.

`model.C = pyo.Set(initialize=range(len(dsatur.C))) # Colors`

model.N = pyo.Set(initialize=nodes) # Nodes

model.E = pyo.Set(initialize=edges]) # Edges

Next, we instantiate our decision variables.

`model.x = pyo.Var(model.N, model.C, within=pyo.Binary)`

model.y = pyo.Var(model.C, within=pyo.Binary)

And then our constraints.

`# Fill every node with some color`

def fill_cstr(model, i):

return sum(model.x[i, :]) == 1# Do not repeat colors on edges and indicator that color is used

def edge_cstr(model, i, j, c):

return model.x[i, c] + model.x[j, c] <= model.y[c]

# Break symmetry by setting a preference order (not required)

def break_symmetry(model, c):

if model.C.first() == c:

return 0 <= model.y[c]

else:

c_prev = model.C.prev(c)

return model.y[c] <= model.y[c_prev]

model.fill_cstr = pyo.Constraint(model.N, rule=fill_cstr)

model.edge_cstr = pyo.Constraint(model.E, model.C, rule=edge_cstr)

model.break_symmetry = pyo.Constraint(model.C, rule=break_symmetry)

You can try and include other symmetry-breaking constraints and see what works better with your available solver. In some experiments I performed, including a preference order using total nodes assigned to a given color has been worse than ignoring it. Possibly due to the solver’s native symmetry-breaking strategies.

`# Break symmetry by setting a preference order with total assignments`

def break_symmetry_agg(model, c):

if model.C.first() == c:

return 0 <= sum(model.x[:, c])

else:

c_prev = model.C.prev(c)

return sum(model.x[:, c]) <= sum(model.x[:, c_prev])

At last, the objective…

`# Total number of colors used`

def obj(model):

return sum(model.y[:])model.obj = pyo.Objective(rule=obj)

And our model is ready to be solved! To do that I’m using the HiGHS persistent solver, which is available in *pyomo* in case *highspy* is also installed in your Python environment.

`solver = pyo.SolverFactory("appsi_highs")`

solver.highs_options["time_limit"] = 120

res = solver.solve(model)

print(res)

For large instances, our solver might have a hard time trying to improve heuristic solutions. However, for a 32-node instance available in the code repository, the solver was able to reduce the number of used colors from 9 to 8. Worth mentioning that it took 24 seconds to complete execution whereas the *DSatur* algorithm for the same instance took only 6 milliseconds (using pure Python, which is an interpreted language).

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