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

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

* test: add failing GraphPartitioning to QUBO coverage

* feat: add GraphPartitioning to QUBO reduction

* docs: add GraphPartitioning to QUBO paper entry

* chore: finalize GraphPartitioning to QUBO pipeline work

* Add GraphPartitioning->QUBO to dominated-rules allow-list

After merging main (which added GraphPartitioning->MaxCut), the
indirect path GraphPartitioning->MaxCut->SpinGlass->QUBO dominates
the direct GraphPartitioning->QUBO reduction.

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

---------

Co-authored-by: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

Author: GiggleLiu

docs/paper/reductions.typ

@@ -4737,6 +4737,48 @@ where $P$ is a penalty weight large enough that any constraint violation costs m
   _Solution extraction._ For each vertex $u$, find terminal position $t$ with $x_(u,t) = 1$. For each edge $(u,v)$, output 1 (cut) if $u$ and $v$ are in different components, 0 otherwise.
 ]
 
+#let gp_qubo = load-example("GraphPartitioning", "QUBO")
+#let gp_qubo_sol = gp_qubo.solutions.at(0)
+#let gp_qubo_edges = gp_qubo.source.instance.graph.edges.map(e => (e.at(0), e.at(1)))
+#let gp_qubo_n = gp_qubo.source.instance.graph.num_vertices
+#let gp_qubo_m = gp_qubo_edges.len()
+#let gp_qubo_penalty = gp_qubo_m + 1
+#let gp_qubo_diag = range(0, gp_qubo_n).map(i => gp_qubo.target.instance.matrix.at(i).at(i))
+#let gp_qubo_cut_edges = gp_qubo_edges.filter(e => gp_qubo_sol.source_config.at(e.at(0)) != gp_qubo_sol.source_config.at(e.at(1)))
+#let gp_qubo_cut_size = gp_qubo_cut_edges.len()
+#reduction-rule("GraphPartitioning", "QUBO",
+  example: true,
+  example-caption: [6-vertex balanced partition instance ($n = #gp_qubo_n$, $|E| = #gp_qubo_m$)],
+  extra: [
+    *Step 1 -- Binary partition variables.* Introduce one binary variable per vertex: $x_i = 0$ means vertex $i$ is in the left block, $x_i = 1$ means it is in the right block. For the canonical instance, this gives $n = #gp_qubo_n$ QUBO variables:
+    $ x_0, x_1, x_2, x_3, x_4, x_5 $
+
+    *Step 2 -- Choose the balance penalty.* The source graph has $m = #gp_qubo_m$ edges, so the construction uses $P = m + 1 = #gp_qubo_penalty$. Any imbalance contributes at least $P$, which is already larger than the maximum possible cut size of any balanced partition.
+
+    *Step 3 -- Fill the QUBO matrix.* The diagonal entries are $Q_(i i) = deg(i) + P(1 - n)$, which evaluates here to $(#gp_qubo_diag.map(str).join(", "))$. For every pair $i < j$, start from $Q_(i j) = 2P = #(2 * gp_qubo_penalty)$, then subtract $2$ when $(i,j)$ is an edge. Hence edge coefficients become $18$ while non-edge coefficients stay $20$; the exported upper-triangular matrix matches the issue example exactly.\
+
+    *Step 4 -- Verify a solution.* The exported witness is $bold(x) = (#gp_qubo_sol.target_config.map(str).join(", "))$, which is also the source partition encoding. The cut edges are #gp_qubo_cut_edges.map(e => "(" + str(e.at(0)) + "," + str(e.at(1)) + ")").join(", "), so the cut size is $#gp_qubo_cut_size$ and the balance penalty vanishes because exactly #(gp_qubo_n / 2) vertices are assigned to the right block #sym.checkmark.
+  ],
+)[
+  Graph Partitioning (minimum bisection) asks for a balanced bipartition minimizing the number of crossing edges. Lucas's Ising formulation @lucas2014 translates directly to a QUBO by combining a cut-counting quadratic objective with a quadratic equality penalty enforcing $sum_i x_i = n / 2$. The reduction uses one binary variable per source vertex, so the QUBO has exactly $n$ variables.
+][
+  _Construction._ Given an undirected graph $G = (V, E)$ with even $n = |V|$ and $m = |E|$, introduce binary variables $x_i in {0,1}$ for each vertex $i in V$. Interpret $x_i = 0$ as $i in A$ and $x_i = 1$ as $i in B$. The cut objective is:
+  $ H_"cut" = sum_((u,v) in E) (x_u + x_v - 2 x_u x_v) $
+  because the term equals $1$ exactly when edge $(u,v)$ crosses the partition. To enforce balance, add:
+  $ H_"bal" = P (sum_i x_i - n/2)^2 $
+  with penalty $P = m + 1$. The QUBO objective is $H = H_"cut" + H_"bal"$.
+
+  Expanding $H_"bal"$ with $x_i^2 = x_i$ gives:
+  $ H_"bal" = P (1 - n) sum_i x_i + 2 P sum_(i < j) x_i x_j + P n^2 / 4 $
+  so the upper-triangular QUBO coefficients are:
+  $ Q_(i i) = deg(i) + P (1 - n) $
+  and for $i < j$, $Q_(i j) = 2 P$ for every pair, then subtract $2$ whenever $(i,j) in E$. Equivalently, edge pairs have coefficient $2P - 2$ and non-edge pairs have coefficient $2P$. The additive constant $P n^2 / 4$ does not affect the minimizer.
+
+  _Correctness._ ($arrow.r.double$) If $bold(x)$ encodes a balanced partition, then $sum_i x_i = n/2$ and $H_"bal" = 0$, so the QUBO objective equals the cut size exactly. ($arrow.l.double$) If $bold(x)$ is imbalanced, then $|sum_i x_i - n/2| >= 1$, hence $H_"bal" >= P = m + 1$. Every balanced partition has cut size at most $m$, so any imbalanced assignment has objective strictly larger than at least one balanced assignment. Therefore every QUBO minimizer is balanced, and among balanced assignments minimizing $H$ is identical to minimizing the cut size.
+
+  _Solution extraction._ Return the QUBO bit-vector directly: the same binary assignment already records the source partition.
+]
+
 #let qubo_ilp = load-example("QUBO", "ILP")
 #let qubo_ilp_sol = qubo_ilp.solutions.at(0)
 #reduction-rule("QUBO", "ILP",

src/rules/graphpartitioning_qubo.rs

@@ -0,0 +1,99 @@
+//! Reduction from GraphPartitioning to QUBO.
+//!
+//! Uses the penalty-method QUBO
+//! H = sum_(u,v in E) (x_u + x_v - 2 x_u x_v) + P (sum_i x_i - n/2)^2
+//! with P = |E| + 1 so any imbalanced partition is dominated by a balanced one.
+
+use crate::models::algebraic::QUBO;
+use crate::models::graph::GraphPartitioning;
+use crate::reduction;
+use crate::rules::traits::{ReduceTo, ReductionResult};
+use crate::topology::{Graph, SimpleGraph};
+
+/// Result of reducing GraphPartitioning to QUBO.
+#[derive(Debug, Clone)]
+pub struct ReductionGraphPartitioningToQUBO {
+    target: QUBO<f64>,
+}
+
+impl ReductionResult for ReductionGraphPartitioningToQUBO {
+    type Source = GraphPartitioning<SimpleGraph>;
+    type Target = QUBO<f64>;
+
+    fn target_problem(&self) -> &Self::Target {
+        &self.target
+    }
+
+    fn extract_solution(&self, target_solution: &[usize]) -> Vec<usize> {
+        target_solution.to_vec()
+    }
+}
+
+#[reduction(overhead = { num_vars = "num_vertices" })]
+impl ReduceTo<QUBO<f64>> for GraphPartitioning<SimpleGraph> {
+    type Result = ReductionGraphPartitioningToQUBO;
+
+    fn reduce_to(&self) -> Self::Result {
+        let n = self.num_vertices();
+        let penalty = self.num_edges() as f64 + 1.0;
+        let mut matrix = vec![vec![0.0f64; n]; n];
+        let mut degrees = vec![0usize; n];
+        let edges = self.graph().edges();
+
+        for &(u, v) in &edges {
+            degrees[u] += 1;
+            degrees[v] += 1;
+        }
+
+        for (i, row) in matrix.iter_mut().enumerate() {
+            row[i] = degrees[i] as f64 + penalty * (1.0 - n as f64);
+            for value in row.iter_mut().skip(i + 1) {
+                *value = 2.0 * penalty;
+            }
+        }
+
+        for (u, v) in edges {
+            let (lo, hi) = if u < v { (u, v) } else { (v, u) };
+            matrix[lo][hi] -= 2.0;
+        }
+
+        ReductionGraphPartitioningToQUBO {
+            target: QUBO::from_matrix(matrix),
+        }
+    }
+}
+
+#[cfg(feature = "example-db")]
+pub(crate) fn canonical_rule_example_specs() -> Vec<crate::example_db::specs::RuleExampleSpec> {
+    use crate::export::SolutionPair;
+
+    vec![crate::example_db::specs::RuleExampleSpec {
+        id: "graphpartitioning_to_qubo",
+        build: || {
+            crate::example_db::specs::rule_example_with_witness::<_, QUBO<f64>>(
+                GraphPartitioning::new(SimpleGraph::new(
+                    6,
+                    vec![
+                        (0, 1),
+                        (0, 2),
+                        (1, 2),
+                        (1, 3),
+                        (2, 3),
+                        (2, 4),
+                        (3, 4),
+                        (3, 5),
+                        (4, 5),
+                    ],
+                )),
+                SolutionPair {
+                    source_config: vec![0, 0, 0, 1, 1, 1],
+                    target_config: vec![0, 0, 0, 1, 1, 1],
+                },
+            )
+        },
+    }]
+}
+
+#[cfg(test)]
+#[path = "../unit_tests/rules/graphpartitioning_qubo.rs"]
+mod tests;

src/rules/mod.rs

@@ -12,6 +12,7 @@ pub(crate) mod coloring_qubo;
 pub(crate) mod factoring_circuit;
 mod graph;
 pub(crate) mod graphpartitioning_maxcut;
+pub(crate) mod graphpartitioning_qubo;
 mod kcoloring_casts;
 mod knapsack_qubo;
 mod ksatisfiability_casts;
@@ -100,6 +101,7 @@ pub(crate) fn canonical_rule_example_specs() -> Vec<crate::example_db::specs::Ru
     specs.extend(coloring_qubo::canonical_rule_example_specs());
     specs.extend(factoring_circuit::canonical_rule_example_specs());
     specs.extend(graphpartitioning_maxcut::canonical_rule_example_specs());
+    specs.extend(graphpartitioning_qubo::canonical_rule_example_specs());
     specs.extend(knapsack_qubo::canonical_rule_example_specs());
     specs.extend(ksatisfiability_qubo::canonical_rule_example_specs());
     specs.extend(ksatisfiability_subsetsum::canonical_rule_example_specs());

src/unit_tests/rules/analysis.rs

@@ -245,6 +245,11 @@ fn test_find_dominated_rules_returns_known_set() {
         ("Factoring", "ILP {variable: \"i32\"}"),
         // K3-SAT → QUBO via SAT → CircuitSAT → SpinGlass chain
         ("KSatisfiability {k: \"K3\"}", "QUBO {weight: \"f64\"}"),
+        // GraphPartitioning -> MaxCut -> SpinGlass -> QUBO is better
+        (
+            "GraphPartitioning {graph: \"SimpleGraph\"}",
+            "QUBO {weight: \"f64\"}",
+        ),
         // Knapsack -> ILP -> QUBO is better than the direct penalty reduction
         ("Knapsack", "QUBO {weight: \"f64\"}"),
         // MaxMatching → MaxSetPacking → ILP is better than direct MaxMatching → ILP

src/unit_tests/rules/graphpartitioning_qubo.rs

@@ -0,0 +1,82 @@
+use super::*;
+use crate::models::algebraic::QUBO;
+use crate::rules::test_helpers::assert_optimization_round_trip_from_optimization_target;
+use crate::topology::SimpleGraph;
+
+fn example_problem() -> GraphPartitioning<SimpleGraph> {
+    GraphPartitioning::new(SimpleGraph::new(
+        6,
+        vec![
+            (0, 1),
+            (0, 2),
+            (1, 2),
+            (1, 3),
+            (2, 3),
+            (2, 4),
+            (3, 4),
+            (3, 5),
+            (4, 5),
+        ],
+    ))
+}
+
+#[test]
+fn test_graphpartitioning_to_qubo_closed_loop() {
+    let source = example_problem();
+    let reduction = ReduceTo::<QUBO<f64>>::reduce_to(&source);
+
+    assert_optimization_round_trip_from_optimization_target(
+        &source,
+        &reduction,
+        "GraphPartitioning->QUBO closed loop",
+    );
+}
+
+#[test]
+fn test_graphpartitioning_to_qubo_matrix_matches_issue_example() {
+    let source = example_problem();
+    let reduction = ReduceTo::<QUBO<f64>>::reduce_to(&source);
+    let qubo = reduction.target_problem();
+
+    assert_eq!(qubo.num_vars(), 6);
+
+    let expected_diagonal = [-48.0, -47.0, -46.0, -46.0, -47.0, -48.0];
+    for (index, expected) in expected_diagonal.into_iter().enumerate() {
+        assert_eq!(qubo.get(index, index), Some(&expected));
+    }
+
+    let edge_pairs = [
+        (0, 1),
+        (0, 2),
+        (1, 2),
+        (1, 3),
+        (2, 3),
+        (2, 4),
+        (3, 4),
+        (3, 5),
+        (4, 5),
+    ];
+    for &(u, v) in &edge_pairs {
+        assert_eq!(qubo.get(u, v), Some(&18.0), "edge ({u}, {v})");
+    }
+
+    let non_edge_pairs = [(0, 3), (0, 4), (0, 5), (1, 4), (1, 5), (2, 5)];
+    for &(u, v) in &non_edge_pairs {
+        assert_eq!(qubo.get(u, v), Some(&20.0), "non-edge ({u}, {v})");
+    }
+}
+
+#[cfg(feature = "example-db")]
+#[test]
+fn test_graphpartitioning_to_qubo_canonical_example_spec() {
+    let spec = canonical_rule_example_specs()
+        .into_iter()
+        .find(|spec| spec.id == "graphpartitioning_to_qubo")
+        .expect("missing canonical GraphPartitioning -> QUBO example spec");
+    let example = (spec.build)();
+
+    assert_eq!(example.source.problem, "GraphPartitioning");
+    assert_eq!(example.target.problem, "QUBO");
+    assert_eq!(example.target.instance["num_vars"], 6);
+    assert!(!example.solutions.is_empty());
+}