To refine our solutions obtained using heuristics and attempt to prove the optimality of the solutions, let us formulate the graph coloring problem as an ILP model. Keep in mind that it might not be able to handle large instances though. The model presented in this section and other exact algorithms are presented in chapter 3 by Lewis (2021).
Let’s define it Sets taken into account in this approach:
- VS: Colors
- NOT: Nodes (or vertices)
- E: Edges
A first question already arises: “How many colors should we consider?” “. In the worst case, all nodes are connected, so one must consider VS the same size as NOT. However, this approach could make our solutions even more difficult 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 get an upper limit for the number of colors.
In this problem, we have two groups of decision variables:
- X_{I, vs}: Are binary variables indicating this node I is colored in vs
- y_{vs}: Are binary variables indicating this color vs is used
We must also create constraints to ensure that:
- Each node must be colored
- If a node on an edge has a color, make sure the color is used
- At most 1 node on each edge can be colored with a given color
- Break the symmetry (this is not required, but it can help)
Finally, our goal should minimize the total number of colors used, which is the sum of Yes. In summary, we have the following equations.
Without further ado, let’s import pyomo for the entire programming model.
import pyomo.environ as pyo
There are two approaches to 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 provided, while, in the second, the model instance is created immediately as its elements are defined. You can learn 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 formulation of the model.
model = pyo.ConcreteModel()
Next, let’s instantiate our Sets. Parse iterables directly from dsatur attributes N And C is valid, we could therefore use them in the initialize keyword arguments. Alternatively, I will forward the original knots And edges from our input data and create a range from the DSatur like 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 including other symmetry breaking constraints and see what works best with your available solver. In some experiments I’ve done, including a preference ordering using the total number of nodes assigned to a given color has proven to be worse than ignoring it. Perhaps 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))
Finally, 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 this, I use the HiGHS persistent solver, available in pyomo in case spying 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 struggle trying to improve heuristic solutions. However, for a 32 node instance available in the code repositorythe solver was able to reduce the number of colors used from 9 to 8. It is worth mentioning that it took 24 seconds to complete the execution while the DSatur The algorithm for the same instance took only 6 milliseconds (using pure Python, which is an interpreted language).
