Solving Partitioning Problem Using Vector of variables

Partitioning problem

Let $w=(w_0, w_1, \ldots, w_{n-1})$ be $n$ positive numbers. The partitioning problem is to partition these numbers into two sets $P$ and $Q$ ($=\overline{P}$) such that the sums of the elements in the two sets are as close as possible. More specifically, the problem is to find a subset $L \subseteq \lbrace 0,1,\ldots, n-1\rbrace$ that minimizes:

\[\begin{aligned} P(L) &= \sum_{i\in L}w_i \\ Q(L) &= \sum_{i\not\in L}w_i \\ f(L) &= \left| P(L)-Q(L) \right| \end{aligned}\]

This problem can be formulated as a QUBO problem. Let $x=(x_0, x_1, \ldots, x_{n-1})$ be binary variables representing the set $L$, that is, $i\in L$ if and only if $x_i=1$. We can rewrite $P(L)$, $Q(L)$ and $f(L)$ using $x$ as follows:

\[\begin{aligned} P(x) &= \sum_{i=0}^{n-1} w_ix_i \\ Q(x) &= \sum_{i=0}^{n-1} w_i (1-x_i) \\ f(x) &= \left( P(x)-Q(x) \right)^2 \end{aligned}\]

Clearly, $f(x)=f(L)^2$ holds. The function $f(x)$ is a quadratic expression of $x$, and an optimal solution that minimizes $f(x)$ also gives an optimal solution to the original partitioning problem.

PyQBPP program for the partitioning problem

The following program creates the QUBO formulation of the partitioning problem for a fixed set of 8 numbers and finds a solution using the Exhaustive Solver.

from pyqbpp import var, expr, sqr, ExhaustiveSolver

w = [64, 27, 47, 74, 12, 83, 63, 40]

x = var("x", len(w))
p = expr()
q = expr()
for i in range(len(w)):
    p += w[i] * x[i]
    q += w[i] * (1 - x[i])
f = sqr(p - q)
print("f =", f.simplify_as_binary())

solver = ExhaustiveSolver(f)
sol = solver.search()

print("Solution:", sol)
print("f(sol) =", sol.eval(f))
print("p(sol) =", sol.eval(p))
print("q(sol) =", sol.eval(q))

print("P :", end="")
for i in range(len(w)):
    if sol.get(x[i]) == 1:
        print(f" {w[i]}", end="")
print()

print("Q :", end="")
for i in range(len(w)):
    if sol.get(x[i]) == 0:
        print(f" {w[i]}", end="")
print()

In this program, w is a Python list with 8 numbers. A vector x of len(w)=8 binary variables is defined. Two Expr objects p and q are defined, and the expressions for $P(x)$ and $Q(x)$ are constructed in the for-loop. An Expr object f stores the expression for $f(x)$.

An Exhaustive Solver object solver for f is created and the solution sol (a Sol object) is obtained by calling its search() method.

The values of $f(x)$, $P(x)$, and $Q(x)$ are evaluated by calling sol.eval(f), sol.eval(p) and sol.eval(q), respectively. The numbers in the sets $L$ and $\overline{L}$ are displayed using the for loops. In these loops, sol.get(x[i]) returns the value of x[i] in sol.

This program outputs:

f = 168100 -88576*x[0] ...
Solution: Sol(energy=0, x[0]=0, x[1]=0, x[2]=1, x[3]=0, x[4]=1, x[5]=1, x[6]=1, x[7]=0)
f(sol) = 0
p(sol) = 205
q(sol) = 205
P : 47 12 83 63
Q : 64 27 74 40

NOTE For a Sol object sol and an Expr object f, sol.eval(f) returns the evaluated value of f on sol. For a Var object a, sol.get(a) returns the value of a in the solution sol.

PyQBPP program using vector operations

PyQBPP has rich vector operations that can simplify the code. Since the overloaded operator * for Vector performs element-wise multiplication, sum(Vector(w) * x) returns the Expr object representing $P(L)$. Furthermore, the overloaded operator - for a scalar and a Vector object returns a Vector object whose components are the scalar minus each element of the vector. Thus, sum(Vector(w) * (1 - x)) returns an Expr object storing $Q(L)$.

from pyqbpp import var, sum, sqr, Vector

w = Vector([64, 27, 47, 74, 12, 83, 63, 40])
x = var("x", len(w))
p = sum(w * x)
q = sum(w * (1 - x))
f = sqr(p - q)

PyQBPP programs can be simplified by using these vector operations.

NOTE The operators +, -, and * are overloaded both for two Vector objects and for a scalar and a Vector object. For two Vector objects, the overloaded operators perform element-wise operations. For a scalar and a Vector object, the overloaded operators apply the scalar operation to each element of the vector.

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