Minimum Graph Bisection Problem

Given an undirected graph $G=(V,E)$ with $n$ nodes (where $n$ is even), the Minimum Graph Bisection problem aims to partition the node set $V$ into two disjoint subsets $S$ and $\overline{S}$ of equal size ($\lvert S\rvert=\lvert\overline{S}\rvert=n/2$) so that the number of edges crossing the partition is minimized.

This problem differs from Max-Cut in two ways:

1. The partition must be balanced (equal-sized halves).
2. We minimize (rather than maximize) the number of crossing edges.

Minimum Graph Bisection is NP-hard and arises in circuit partitioning, parallel computing, and graph-based data clustering.

QUBO Formulation

Assume that the nodes are labeled $0,1,\ldots,n-1$. We introduce $n$ binary variables $x_0, x_1, \ldots, x_{n-1}$, where $x_i=1$ if and only if node $i$ belongs to $S$.

Objective

The number of edges crossing the partition is:

\[\text{objective} = \sum_{(i,j)\in E}\Bigl(x_i(1-x_j) + (1-x_i)x_j\Bigr)\]

We want to minimize this value.

Constraint

The partition must be balanced:

\[\text{constraint} = \Bigl(\sum_{i=0}^{n-1} x_i = \frac{n}{2}\Bigr)\]

This constraint expression equals 0 when satisfied.

QUBO expression

The final QUBO expression combines the objective and constraint with a penalty weight $P$:

\[f = \text{objective} + P \times \text{constraint}\]

where $P$ must be large enough (e.g., $P = \lvert E\rvert + 1$) to ensure that the balance constraint is always satisfied in an optimal solution.

PyQBPP program

The following PyQBPP program solves the Minimum Graph Bisection problem for a 16-node graph:

import pyqbpp as qbpp

N = 16
edges = [
    (0, 1),   (0, 2),   (1, 3),   (1, 4),   (2, 5),   (2, 6),   (3, 7),
    (3, 13),  (4, 6),   (4, 7),   (4, 14),  (5, 8),   (6, 8),   (6, 12),
    (6, 14),  (7, 14),  (8, 9),   (9, 10),  (9, 12),  (10, 11), (10, 12),
    (11, 13), (11, 15), (12, 14), (12, 15), (13, 15), (14, 15),
]
M = len(edges)

x = qbpp.var("x", N)

# Objective: number of edges crossing the cut
objective = 0
for i, j in edges:
    objective += x[i] * ~x[j] + ~x[i] * x[j]

# Constraint: exactly N/2 nodes in each partition
constraint = qbpp.sum(x) == N // 2

# Penalty weight: M + 1 ensures constraint is prioritized
f = objective + (M + 1) * constraint
f.simplify_as_binary()

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

print(f"Cut edges = {sol(objective)}")
print(f"constraint = {sol(constraint)}")

In this program, the objective counts the number of edges crossing the cut, and the constraint enforces that exactly $N/2$ nodes are in each partition. The penalty weight $P = M + 1$ ensures that the balance constraint is always satisfied. Unlike the Max-Cut problem where we negate the objective for maximization, here we minimize the objective directly.

Output

Cut edges = 6
constraint = 0

The solver finds a balanced partition with only 6 edges crossing the cut.