Solving Partitioning Problem Using Array 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:

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

where $\overline{x_i}$ denotes the negated literal of $x_i$, which takes the value $1$ when $x_i=0$ and $0$ when $x_i=1$. Mathematically, $\overline{x_i} = 1 - x_i$, but QUBO++ handles negated literals natively using the ~ operator (e.g., ~x[i]), which avoids expanding $1 - x_i$ and is more efficient. For more details, see Negated Literals.

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.

QUBO++ program for the partitioning problem

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

#include <qbpp/qbpp.hpp>
#include <qbpp/exhaustive_solver.hpp>

int main() {
  std::vector<uint32_t> w = {64, 27, 47, 74, 12, 83, 63, 40};

  auto x = qbpp::var("x", w.size());
  auto p = qbpp::expr();
  auto q = qbpp::expr();
  for (size_t i = 0; i < w.size(); ++i) {
    p += w[i] * x[i];
    q += w[i] * ~x[i];
  }
  auto f = qbpp::sqr(p - q);
  std::cout << "f = " << f.simplify_as_binary() << std::endl;

  auto solver = qbpp::exhaustive_solver::ExhaustiveSolver(f);
  auto sol = solver.search();

  std::cout << "Solution: " << sol << std::endl;
  std::cout << "f(sol) = " << f(sol) << std::endl;
  std::cout << "p(sol) = " << p(sol) << std::endl;
  std::cout << "q(sol) = " << q(sol) << std::endl;

  std::cout << "P :";
  for (size_t i = 0; i < w.size(); ++i) {
    if (sol(x[i]) == 1) {
      std::cout << " " << w[i];
    }
  }
  std::cout << std::endl;

  std::cout << "Q :";
  for (size_t i = 0; i < w.size(); ++i) {
    if (sol(x[i]) == 0) {
      std::cout << " " << w[i];
    }
  }
  std::cout << std::endl;
}

In this program, w is defined as a std::vector object with 8 numbers. An array x of w.size()=8 binary variables is defined. Two qbpp::Expr objects p and q are defined, and the expressions for $P(x)$ and $Q(x)$ are constructed in the for-loop. Here, ~x[i] denotes the negated literal $\overline{x_i}$ of x[i]. A qbpp::Expr object f stores the expression for $f(x)$.

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

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

This program outputs:

f = 168100 -88576*x[0] -41364*x[1] -68244*x[2] -99456*x[3] -19104*x[4] -108564*x[5] -87444*x[6] -59200*x[7] +13824*x[0]*x[1] +24064*x[0]*x[2] +37888*x[0]*x[3] +6144*x[0]*x[4] +42496*x[0]*x[5] +32256*x[0]*x[6] +20480*x[0]*x[7] +10152*x[1]*x[2] +15984*x[1]*x[3] +2592*x[1]*x[4] +17928*x[1]*x[5] +13608*x[1]*x[6] +8640*x[1]*x[7] +27824*x[2]*x[3] +4512*x[2]*x[4] +31208*x[2]*x[5] +23688*x[2]*x[6] +15040*x[2]*x[7] +7104*x[3]*x[4] +49136*x[3]*x[5] +37296*x[3]*x[6] +23680*x[3]*x[7] +7968*x[4]*x[5] +6048*x[4]*x[6] +3840*x[4]*x[7] +41832*x[5]*x[6] +26560*x[5]*x[7] +20160*x[6]*x[7]
Solution: 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 qbpp::Expr object f and a qbpp::Sol object sol, both f(sol) and sol(f) return the resulting value of f evaluated on sol. Likewise, for a qbpp::Var object a, both a(sol) and sol(a) return the value of a in the solution sol. The form f(sol) is natural from a mathematical perspective, as it corresponds to evaluating a function at a point. In contrast, sol(f) is natural from an object-oriented programming perspective, where the solution object evaluates an expression. You may use either form according to your preference.

QUBO++ program using array operations

QUBO++ has rich array operations that can simplify the code. In the following code, w is defined using qbpp::int_array(), and x is an array of binary variables created by qbpp::var(). Since the overloaded operator * for qbpp::Array returns the element-wise product, qbpp::sum(w * x) returns the qbpp::Expr object representing $P(L)$. The ~ operator applied to an array of variables returns an array of their negated literals. Thus, qbpp::sum(w * ~x) returns a qbpp::Expr object storing $Q(L)$.

  auto w = qbpp::int_array({64, 27, 47, 74, 12, 83, 63, 40});
  auto x = qbpp::var("x", w.size());
  auto p = qbpp::sum(w * x);
  auto q = qbpp::sum(w * ~x);
  auto f = qbpp::sqr(p - q);

QUBO++ programs can be simplified by using these array operations. In addition, since array operations for large arrays are parallelized by multithreading, they can accelerate the process of creating QUBO models.

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