[Rule] GraphPartitioning to QUBO

Source: GraphPartitioning
Target: QUBO
Motivation: Enables solving graph bisection on quantum annealers (D-Wave); natural quadratic formulation with balance penalty.
Reference: Lucas, 2014, Ising formulations of many NP problems, Section 2.2

Reduction Algorithm

Notation:

Source: undirected graph $G = (V, E)$, $n = |V|$ (even), $m = |E|$
Target: QUBO with $n$ binary variables

Variable mapping: $x_i \in {0, 1}$ for each vertex $i$ ($x_i = 0$ if $i \in A$, $x_i = 1$ if $i \in B$).

Step 1 — Cut-counting term. For each edge $(u, v) \in E$, the expression $x_u + x_v - 2x_u x_v$ equals 1 iff $x_u \neq x_v$ (i.e., the edge is cut). Summing over all edges counts the cut size.

Step 2 — Balance penalty. The quadratic penalty $P(\sum_{i \in V} x_i - n/2)^2$ is zero iff exactly $n/2$ vertices are in $B$. Set $P > m$ to ensure any imbalanced partition has higher cost than any balanced one.

Step 3 — Construct the QUBO Q matrix. Initialize an $n \times n$ upper-triangular matrix $Q$ to zero. For each edge $(u,v) \in E$ with $u < v$:

$Q_{uu} \mathrel{+}= 1$, $Q_{vv} \mathrel{+}= 1$ (linear cut terms)
$Q_{uv} \mathrel{+}= -2$ (quadratic cut term)

Then add the expanded penalty $P(\sum x_i - n/2)^2 = P(1-n)\sum_i x_i + 2P\sum_{i<j} x_i x_j + Pn^2/4$:

For each $i$: $Q_{ii} \mathrel{+}= P(1-n)$
For each $i < j$: $Q_{ij} \mathrel{+}= 2P$

Resulting QUBO Q matrix entries:

$$Q_{ii} = \deg(i) + P(1 - n)$$

$$Q_{ij} = \begin{cases} -2 + 2P & \text{if } (i,j) \in E \\ 2P & \text{if } (i,j) \notin E \end{cases} \quad (i < j)$$

The QUBO objective is $\min_{x \in {0,1}^n} x^\top Q x$, equivalent to the original up to the constant offset $Pn^2/4$.

Solution extraction: $A = {i : x_i = 0}$, $B = {i : x_i = 1}$.

Size Overhead

Target metric (code name)	Polynomial (using symbols above)
`num_vars`	$n$

Validation Method

Closed-loop testing: solve GraphPartitioning by brute-force, solve the reduced QUBO, and verify both give the same optimal cut. Verify penalty $P$ is large enough.

Example

Source: 6 vertices, 9 edges: $(0,1), (0,2), (1,2), (1,3), (2,3), (2,4), (3,4), (3,5), (4,5)$.

QUBO: 6 variables, penalty $P = 10 > m = 9$.

Degrees: $\deg(0)=2, \deg(1)=3, \deg(2)=4, \deg(3)=4, \deg(4)=3, \deg(5)=2$.

$Q_{ii} = \deg(i) + 10(1-6) = \deg(i) - 50$:

$$Q_{ii} = (-48,\ -47,\ -46,\ -46,\ -47,\ -48)$$

Off-diagonal ($i < j$): $-2 + 2 \cdot 10 = 18$ for edges, $2 \cdot 10 = 20$ for non-edges:

$$Q = \begin{pmatrix} -48 & 18 & 18 & 20 & 20 & 20 \\ & -47 & 18 & 18 & 20 & 20 \\ & & -46 & 18 & 18 & 20 \\ & & & -46 & 18 & 18 \\ & & & & -47 & 18 \\ & & & & & -48 \end{pmatrix}$$

Optimal: $x = (0,0,0,1,1,1)$, cut edges: $(1,3), (2,3), (2,4)$, cut $= 3$, balance: $|B| = 3 = n/2$. Penalty $= 0$, $H = 3$.

Infeasible example: $x = (0,0,0,0,1,1)$ ($|B| = 2 \neq 3$), cut $= 3$ (edges $(2,4), (3,4), (3,5)$), penalty $= 10 \cdot (2 - 3)^2 = 10$, $H = 3 + 10 = 13$.

Issue Quality Check — Rule

Check	Result	Details
Usefulness	⚠️ Warn	Source "GraphPartitioning" not yet in codebase
Non-trivial	✅ Pass	Penalty-weighted QUBO construction
Correctness	⚠️ Warn	Reference is real but wrong section cited
Well-written	❌ Fail	Example contains arithmetic errors

Overall: 1 passed, 1 failed, 2 warnings

Usefulness

GraphPartitioning does not exist in the codebase. Requires a [Model] issue first.

Non-trivial

Constructs a QUBO objective with cut-counting quadratic term and balance-enforcing penalty. Standard but non-trivial encoding.

Correctness — WARN

Lucas (2014) is a verified reference (already in bibliography as lucas2014). However, the issue cites "Section 5.2" — the graph partitioning formulation is actually in Section 2.2 of the paper.

The QUBO formulation H = Σ(xᵤ + x_v − 2xᵤx_v) + P(Σxᵢ − n/2)² is mathematically correct. Setting P > m ensures balance.

Well-written — FAIL

Example arithmetic errors:

The infeasible example x = (0,0,0,0,1,1) claims:

"cut = 2" — incorrect. Checking edges: (2,4), (3,4), (3,5) are all cut edges → cut = 3
"H = 12" — incorrect. With cut = 3 and penalty = 10·(2−3)² = 10: H = 3 + 10 = 13

The optimal example (x = (0,0,0,1,1,1), cut = 3, H = 3) is correct.

Required fixes

Correct section reference from "Section 5.2" → Section 2.2
Fix infeasible example: cut = 3 (not 2), H = 13 (not 12)
File/link a [Model] issue for GraphPartitioning

Issue Quality Check — Rule

Check	Result	Details
Usefulness	✅ Pass	Source problem `GraphPartitioning` does not yet exist in the reduction graph; novel reduction
Non-trivial	✅ Pass	Constructs quadratic penalty terms and cut-counting objective; genuine structural transformation
Correctness	❌ Fail	Section reference is wrong: the issue cites Section 5.2 of Lucas (2014), but that section covers "Knapsack with Integer Weights" — graph partitioning is in Section 2.2
Well-written	⚠️ Warn	Algorithm gives Hamiltonian formula but does not explicitly show how to populate the QUBO Q matrix entries; some implementability gaps

Overall: 2 passed, 1 failed, 1 warning

Usefulness

The source problem GraphPartitioning does not exist in the codebase (pred show GraphPartitioning fails). Since neither the problem nor any reduction from it exists, this is a novel contribution. However, note that the source model must be implemented first before this rule can be added.

Non-trivial

The reduction constructs a QUBO objective with two distinct components:

A cut-counting term $\sum_{(u,v) \in E}(x_u + x_v - 2x_u x_v)$ that counts edges between partitions
A quadratic balance penalty $P(\sum x_i - n/2)^2$ that enforces equal partition sizes

This involves genuine structural transformation — encoding graph structure and balance constraints into a single unconstrained quadratic objective with penalty terms. Pass.

Correctness

Reference verification:

The issue cites: Lucas, 2014, "Ising formulations of many NP problems" (DOI: 10.3389/fphy.2014.00005), Section 5.2.

✅ The paper exists and is correctly identified — published in Frontiers in Physics, February 2014, by Andrew Lucas. It is already in the project bibliography (references.bib entry lucas2014).
❌ Wrong section number. Section 5.2 of Lucas (2014) covers "Knapsack with Integer Weights" (Equations 47–50). The graph partitioning formulation is in Section 2.2, "Graph Partitioning" (Equations 7–10).

Formulation verification:

Despite the wrong section citation, the QUBO formulation in the issue is mathematically correct. Lucas (2014) Section 2.2 uses Ising spins $s_v = \pm 1$ with:

$H_A = A(\sum s_i)^2$ (balance penalty)
$H_B = B \sum_{(uv) \in E} \frac{1 - s_u s_v}{2}$ (cut edges)

The issue's formulation uses binary variables $x_i \in {0,1}$ (related by $x = (s+1)/2$), which is the standard QUBO conversion. The resulting Hamiltonian $H = \sum_{(u,v)}(x_u + x_v - 2x_u x_v) + P(\sum x_i - n/2)^2$ is equivalent after this change of variables.

The penalty condition $P > m$ is sufficient to enforce balance, consistent with the paper's analysis (Equation 10).

Well-written

Template completeness: All required sections are present (Source, Target, Motivation, Reference, Reduction Algorithm, Size Overhead, Validation Method, Example). ✅

Symbol consistency: Symbols $n$, $m$, $G$, $V$, $E$, $P$ are all defined before use. ✅

Overhead table: num_vars = n is correct — the QUBO has exactly $n$ binary variables (one per vertex). The target's size_fields confirms num_vars is the correct field name. ✅

Algorithm completeness (⚠️ Warning):
The Hamiltonian is clearly specified, but the issue does not explicitly describe how to construct the QUBO Q matrix entries ($Q_{ij}$) from the Hamiltonian. An implementer needs to:

Expand the balance penalty $P(\sum x_i - n/2)^2$ into individual quadratic and linear terms
Map all terms to the upper-triangular Q matrix: diagonal entries $Q_{ii}$ and off-diagonal entries $Q_{ij}$

This step is non-trivial (the penalty expansion produces $O(n^2)$ off-diagonal terms) and should be spelled out for implementability.

Example quality: The 6-vertex example is well-chosen — small enough to verify by hand, includes both a feasible optimal solution and an infeasible configuration showing the penalty in action. ✅

Recommendations

Fix the section reference: Change "Section 5.2" to "Section 2.2" in the Reference field.
Add explicit Q matrix construction: Show how to expand the Hamiltonian into QUBO matrix entries. Specifically:
- Diagonal: $Q_{ii} = \deg(i) + P(1 - n)$ (from cut term + penalty expansion, after collecting linear terms)
- Off-diagonal for $(i,j) \in E$: $Q_{ij} = -2 + 2P$ (cut term + penalty)
- Off-diagonal for $(i,j) \notin E$: $Q_{ij} = 2P$ (penalty only)
- Constant offset: $Pn^2/4$ (can be dropped for optimization)
Consider mentioning that the source model GraphPartitioning must be implemented first (no open issue found for it).

Issue Quality Check — Rule

Check	Result	Details
Usefulness	✅ Pass	No existing path from GraphPartitioning to QUBO; novel reduction
Non-trivial	✅ Pass	Constructs quadratic penalty terms with cut-counting and balance enforcement
Correctness	❌ Fail	Wrong section citation; infeasible example has arithmetic errors
Well-written	⚠️ Warn	Algorithm omits explicit Q matrix construction; example search space slightly small

Overall: 2 passed, 1 failed, 1 warning

Usefulness

pred path GraphPartitioning QUBO returns no path — this is a novel reduction. GraphPartitioning currently has no outgoing reductions at all, so this would be the first connection from the graph partitioning family to QUBO/quantum annealing problems. Pass.

Non-trivial

The reduction constructs a QUBO objective with two distinct components:

A cut-counting term $\sum_{(u,v) \in E}(x_u + x_v - 2x_u x_v)$ that counts edges between partitions
A quadratic balance penalty $P(\sum x_i - n/2)^2$ that enforces equal partition sizes

This involves genuine structural transformation — encoding graph structure and a hard balance constraint into a single unconstrained quadratic objective with penalty terms. Pass.

Correctness

Reference verification:

✅ Lucas (2014), "Ising formulations of many NP problems" (DOI: 10.3389/fphy.2014.00005) — exists, already in bibliography as lucas2014.
❌ Wrong section citation. The issue cites Section 5.2, which is "Knapsack with Integer Weights" (page 14). The graph partitioning formulation is in Section 2.2 (page 6, Equations 7–9).

Formulation verification:

Lucas (2014) Section 2.2 uses Ising spins $s_v = \pm 1$ with $H_A = A(\sum s_i)^2$ (balance) and $H_B = B \sum_{(uv)} (1 - s_u s_v)/2$ (cut). Under the standard substitution $s_i = 1 - 2x_i$, this becomes exactly the issue's QUBO formula. ✅

Penalty condition: $P > m$ is sufficient to enforce balance. ✅

Overhead verification: num_vars = n maps to num_vars = num_vertices in code. GraphPartitioning has num_vertices() getter; QUBO has num_vars as its size field. The QUBO has exactly $n$ binary variables (one per vertex). ✅

Infeasible example errors:

The infeasible example $x = (0,0,0,0,1,1)$ (partition B = {4,5}) claims "cut = 2, H = 12". Verified by enumeration:

Cut edges crossing the partition: $(2,4), (3,4), (3,5)$ → cut = 3 (not 2)
Balance penalty: $10 \cdot (2 - 3)^2 = 10$
H = 3 + 10 = 13 (not 12)

The optimal example $x = (0,0,0,1,1,1)$ with $H = 3$ is correct (confirmed by brute-force enumeration of all 64 configurations; tied with complement $(1,1,1,0,0,0)$).

Well-written

Required sections: Source ✅, Target ✅, Motivation ✅, Reference ✅, Reduction Algorithm ✅, Size Overhead ✅, Validation Method ✅, Example ✅ — all present and substantive.

Symbol consistency: Symbols $n$, $m$, $G$, $V$, $E$, $P$ are defined before use. Overhead metric num_vars matches QUBO's size field. ✅

Algorithm completeness (⚠️ Warning): The Hamiltonian formula is clearly specified, but the issue does not show how to populate the QUBO Q matrix entries from the expanded Hamiltonian. An implementer needs to:

Expand the penalty $P(\sum x_i - n/2)^2$ into individual quadratic and linear terms
Map all terms to the upper-triangular Q matrix: diagonal $Q_{ii}$ and off-diagonal $Q_{ij}$

This is non-trivial (the penalty expansion produces $O(n^2)$ off-diagonal terms).

Example quality (⚠️ Warning): Search space = $2^6 = 64$, below the recommended 128. There are 20 feasible balanced partitions and 18 suboptimal ones, which is sufficient for testing diversity. Recommend using 7–8 vertices for a stronger example.

Required fixes

Correctness: Change section reference from "Section 5.2" → Section 2.2
Correctness: Fix infeasible example: cut = 3 (not 2), H = 13 (not 12)

Recommendations

Add explicit Q matrix construction showing how to compute $Q_{ii}$ and $Q_{ij}$ entries
Consider a slightly larger example (7–8 vertices) for search space ≥ 128

Fix-issue changelog

Reference: Changed "Section 5.2" → "Section 2.2" (Section 5.2 is Knapsack with Integer Weights; graph partitioning is in Section 2.2)
Algorithm: Added Step 3 with explicit Q matrix construction procedure and closed-form Q matrix entries
Example: Fixed infeasible example arithmetic: cut = 3 (not 2), H = 13 (not 12)
Example: Added explicit Q matrix, degree table, and diagonal/off-diagonal computation for the 6-vertex instance
Example: Added cut edge enumeration for optimal solution

Applied by /fix-issue.

Add plan for #119: [Rule] GraphPartitioning to QUBO

Please kindly check the following items (PR #706):

Paper (PDF): check definition, proof sketch, and example figure

CLI demo (build from source: cargo install --path problemreductions-cli):

pred show GraphPartitioning
pred create --example GraphPartitioning -o instance.json
pred reduce instance.json GraphPartitioning QUBO -o reduced.json
pred solve reduced.json QUBO

Implementation (Optional): spot-check the source files changed in this PR for correctness

💬 Join the discussion on Zulip — feel free to ask questions or leave feedback there.

Fix #119: [Rule] GraphPartitioning to QUBO (#706)

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[Rule] GraphPartitioning to QUBO #119

Reduction Algorithm

Size Overhead

Validation Method

Example

Issue Quality Check — Rule

Usefulness

Non-trivial

Correctness — WARN

Well-written — FAIL

Required fixes

Issue Quality Check — Rule

Usefulness

Non-trivial

Correctness

Well-written

Recommendations

Issue Quality Check — Rule

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Fix-issue changelog

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[Rule] GraphPartitioning to QUBO #119

Description

Reduction Algorithm

Size Overhead

Validation Method

Example

Activity

zazabap commented on Mar 8, 2026

Issue Quality Check — Rule

Usefulness

Non-trivial

Correctness — WARN

Well-written — FAIL

Required fixes

zazabap commented on Mar 11, 2026

Issue Quality Check — Rule

Usefulness

Non-trivial

Correctness

Well-written

Recommendations

isPANN commented on Mar 20, 2026

Issue Quality Check — Rule

Usefulness

Non-trivial

Correctness

Well-written

Required fixes

Recommendations

isPANN commented on Mar 20, 2026

Fix-issue changelog

GiggleLiu commented on Mar 21, 2026

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