Integer Linear Programming (ILP)

Integer Linear Programming (ILP) can be converted into a QUBO expression using PyQBPP. As an example, consider the following ILP:

\[\begin{aligned} \text{Maximize:} && 2x_0 +5x_1+5x_2\\ \text{Subject to:} && x_0 + 3 x_1 + x_2 &\leq 12 \\ && x_0 + 2x_2 &\leq 5\\ && x_1 + x_2 &\leq 4; \end{aligned}\]

PyQBPP program

The following PyQBPP program formulates this ILP as a QUBO expression and solves it using the Easy Solver:

import pyqbpp as qbpp

x = qbpp.var("x", shape=3, between=(0, 5))
objective = 2 * x[0] + 5 * x[1] + 5 * x[2]
c1 = (0 <= x[0] + 3 * x[1] + x[2]) & (qbpp.same <= 12)
c2 = (0 <= x[0] + 2 * x[2]) & (qbpp.same <= 5)
c3 = (0 <= x[1] + x[2]) & (qbpp.same <= 4)

f = -objective + 100 * (c1 + c2 + c3)
f.simplify_as_binary()
solver = qbpp.EasySolver(f)
sol = solver.search(time_limit=1.0)
print(f"x0 = {sol(x[0])}, x1 = {sol(x[1])}, x2 = {sol(x[2])}")
print(f"objective = {sol(objective)}")
print(f"c1 = {sol(c1.body)}, c2 = {sol(c2.body)}, c3 = {sol(c3.body)}")

In this program, x is a vector of three integer variables, each taking an integer value in the range $[0, 5]$. The objective function and the three constraints are represented by objective, c1, c2, and c3, respectively. They are combined into a single QUBO expression f, where the penalty constant 100 is used to enforce the constraints.

The Easy Solver searches for a low-energy solution of f and returns it as sol. The obtained solution and the values of objective, c1.body, c2.body, and c3.body are printed as follows:

x0 = 2, x1 = 3, x2 = 1
objective = 24
c1 = 12, c2 = 4, c3 = 4

We observe that an obtained solution with the objective 24 satisfies all constraints.