Fix #202: [Rule] Partition to Knapsack (#744)

* Add plan for #202: [Rule] Partition to Knapsack

* Implement #202: [Rule] Partition to Knapsack

* chore: remove plan file after implementation

Author: GiggleLiu

docs/paper/reductions.typ

@@ -5929,6 +5929,48 @@ where $P$ is a penalty weight large enough that any constraint violation costs m
   _Solution extraction._ Discard slack variables: return $bold(x)' [0..n]$.
 ]
 
+#let part_ks = load-example("Partition", "Knapsack")
+#let part_ks_sol = part_ks.solutions.at(0)
+#let part_ks_sizes = part_ks.source.instance.sizes
+#let part_ks_n = part_ks_sizes.len()
+#let part_ks_total = part_ks_sizes.fold(0, (a, b) => a + b)
+#let part_ks_capacity = part_ks.target.instance.capacity
+#let part_ks_selected = part_ks_sol.source_config.enumerate().filter(((i, x)) => x == 1).map(((i, x)) => i)
+#let part_ks_selected_sizes = part_ks_selected.map(i => part_ks_sizes.at(i))
+#let part_ks_selected_sum = part_ks_selected_sizes.fold(0, (a, b) => a + b)
+#reduction-rule("Partition", "Knapsack",
+  example: true,
+  example-caption: [#part_ks_n elements, total sum $S = #part_ks_total$],
+  extra: [
+    #pred-commands(
+      "pred create --example " + problem-spec(part_ks.source) + " -o partition.json",
+      "pred reduce partition.json --to " + target-spec(part_ks) + " -o bundle.json",
+      "pred solve bundle.json",
+      "pred evaluate partition.json --config " + part_ks_sol.source_config.map(str).join(","),
+    )
+
+    *Step 1 -- Source instance.* The canonical Partition instance has sizes $(#part_ks_sizes.map(str).join(", "))$ with total sum $S = #part_ks_total$, so a balanced witness must hit exactly $S / 2 = #part_ks_capacity$.
+
+    *Step 2 -- Build the knapsack instance.* The reduction copies each size into both the weight and the value list, producing weights $(#part_ks.target.instance.weights.map(str).join(", "))$, values $(#part_ks.target.instance.values.map(str).join(", "))$, and capacity $C = #part_ks_capacity$. No auxiliary variables are introduced, so the target has the same $#part_ks_n$ binary coordinates as the source.
+
+    *Step 3 -- Verify the canonical witness.* The serialized witness uses the same binary vector on both sides, $bold(x) = (#part_ks_sol.source_config.map(str).join(", "))$. It selects elements at indices $\{#part_ks_selected.map(str).join(", ")\}$ with sizes $(#part_ks_selected_sizes.map(str).join(", "))$, so the chosen subset has total weight and value $#part_ks_selected_sum = #part_ks_capacity$. Hence the knapsack solution saturates the capacity and certifies a balanced partition.
+
+    *Witness semantics.* The example DB stores one canonical balanced subset. This instance has multiple balanced partitions because several different subsets sum to $#part_ks_capacity$, but one witness is enough to demonstrate the reduction.
+  ],
+)[
+  This $O(n)$ reduction#footnote[The linear-time bound follows from a single pass that copies the source sizes into item weights and values.] @garey1979[MP9] constructs a 0-1 Knapsack instance by copying each Partition size into both the item weight and item value and setting the capacity to half the total size sum. For $n$ source elements it produces $n$ knapsack items.
+][
+  _Construction._ Given positive sizes $s_0, dots, s_(n-1)$ with total sum $S = sum_(i=0)^(n-1) s_i$, create one knapsack item per element and set
+  $ w_i = s_i, quad v_i = s_i $
+  for every $i in {0, dots, n-1}$. Set the knapsack capacity to
+  $ C = floor(S / 2). $
+  Every feasible knapsack solution is therefore a subset of the original elements, and because $w_i = v_i$, its objective value equals the same subset sum.
+
+  _Correctness._ ($arrow.r.double$) If the Partition instance is satisfiable, some subset $A'$ has sum $S / 2$. In particular $S$ is even, so $C = S / 2$, and selecting exactly the corresponding knapsack items is feasible with value $S / 2$. No feasible knapsack solution can have value larger than $C$, because value equals weight for every item and total weight is bounded by $C$. Thus the knapsack optimum is exactly $S / 2$. ($arrow.l.double$) If the knapsack optimum is $S / 2$, then the optimum is an integer and hence $S$ must be even. The selected items have total value $S / 2$, so they also have total weight $S / 2$ because $w_i = v_i$ itemwise. Those items therefore form a subset of the original multiset whose complement has the same sum, giving a valid balanced partition.
+
+  _Solution extraction._ Return the same binary selection vector on the original elements: item $i$ is selected in the knapsack witness if and only if element $i$ belongs to the extracted partition subset.
+]
+
 #let ks_qubo = load-example("Knapsack", "QUBO")
 #let ks_qubo_sol = ks_qubo.solutions.at(0)
 #let ks_qubo_num_items = ks_qubo.source.instance.weights.len()

src/rules/mod.rs

@@ -31,6 +31,7 @@ pub(crate) mod minimummultiwaycut_qubo;
 pub(crate) mod minimumvertexcover_maximumindependentset;
 pub(crate) mod minimumvertexcover_minimumfeedbackvertexset;
 pub(crate) mod minimumvertexcover_minimumsetcovering;
+pub(crate) mod partition_knapsack;
 pub(crate) mod sat_circuitsat;
 pub(crate) mod sat_coloring;
 pub(crate) mod sat_ksat;
@@ -114,6 +115,7 @@ pub(crate) fn canonical_rule_example_specs() -> Vec<crate::example_db::specs::Ru
     specs.extend(maximummatching_maximumsetpacking::canonical_rule_example_specs());
     specs.extend(maximumsetpacking_qubo::canonical_rule_example_specs());
     specs.extend(minimummultiwaycut_qubo::canonical_rule_example_specs());
+    specs.extend(partition_knapsack::canonical_rule_example_specs());
     specs.extend(minimumvertexcover_maximumindependentset::canonical_rule_example_specs());
     specs.extend(minimumvertexcover_minimumfeedbackvertexset::canonical_rule_example_specs());
     specs.extend(minimumvertexcover_minimumsetcovering::canonical_rule_example_specs());

src/rules/partition_knapsack.rs

@@ -0,0 +1,73 @@
+//! Reduction from Partition to Knapsack.
+
+use crate::models::misc::{Knapsack, Partition};
+use crate::reduction;
+use crate::rules::traits::{ReduceTo, ReductionResult};
+
+/// Result of reducing Partition to Knapsack.
+#[derive(Debug, Clone)]
+pub struct ReductionPartitionToKnapsack {
+    target: Knapsack,
+}
+
+impl ReductionResult for ReductionPartitionToKnapsack {
+    type Source = Partition;
+    type Target = Knapsack;
+
+    fn target_problem(&self) -> &Self::Target {
+        &self.target
+    }
+
+    fn extract_solution(&self, target_solution: &[usize]) -> Vec<usize> {
+        target_solution.to_vec()
+    }
+}
+
+fn partition_size_to_i64(value: u64) -> i64 {
+    i64::try_from(value)
+        .expect("Partition -> Knapsack requires all sizes and total_sum / 2 to fit in i64")
+}
+
+#[reduction(overhead = {
+    num_items = "num_elements",
+})]
+impl ReduceTo<Knapsack> for Partition {
+    type Result = ReductionPartitionToKnapsack;
+
+    fn reduce_to(&self) -> Self::Result {
+        let weights: Vec<i64> = self
+            .sizes()
+            .iter()
+            .copied()
+            .map(partition_size_to_i64)
+            .collect();
+        let values = weights.clone();
+        let capacity = partition_size_to_i64(self.total_sum() / 2);
+
+        ReductionPartitionToKnapsack {
+            target: Knapsack::new(weights, values, capacity),
+        }
+    }
+}
+
+#[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: "partition_to_knapsack",
+        build: || {
+            crate::example_db::specs::rule_example_with_witness::<_, Knapsack>(
+                Partition::new(vec![3, 1, 1, 2, 2, 1]),
+                SolutionPair {
+                    source_config: vec![1, 0, 0, 1, 0, 0],
+                    target_config: vec![1, 0, 0, 1, 0, 0],
+                },
+            )
+        },
+    }]
+}
+
+#[cfg(test)]
+#[path = "../unit_tests/rules/partition_knapsack.rs"]
+mod tests;

src/unit_tests/rules/partition_knapsack.rs

@@ -0,0 +1,54 @@
+use super::*;
+use crate::models::misc::Partition;
+use crate::rules::test_helpers::assert_satisfaction_round_trip_from_optimization_target;
+use crate::solvers::{BruteForce, Solver};
+use crate::traits::Problem;
+use crate::types::SolutionSize;
+
+#[test]
+fn test_partition_to_knapsack_closed_loop() {
+    let source = Partition::new(vec![3, 1, 1, 2, 2, 1]);
+    let reduction = ReduceTo::<Knapsack>::reduce_to(&source);
+
+    assert_satisfaction_round_trip_from_optimization_target(
+        &source,
+        &reduction,
+        "Partition -> Knapsack closed loop",
+    );
+}
+
+#[test]
+fn test_partition_to_knapsack_structure() {
+    let source = Partition::new(vec![3, 1, 1, 2, 2, 1]);
+    let reduction = ReduceTo::<Knapsack>::reduce_to(&source);
+    let target = reduction.target_problem();
+
+    assert_eq!(target.weights(), &[3, 1, 1, 2, 2, 1]);
+    assert_eq!(target.values(), &[3, 1, 1, 2, 2, 1]);
+    assert_eq!(target.capacity(), 5);
+    assert_eq!(target.num_items(), source.num_elements());
+}
+
+#[test]
+fn test_partition_to_knapsack_odd_total_is_not_satisfying() {
+    let source = Partition::new(vec![2, 4, 5]);
+    let reduction = ReduceTo::<Knapsack>::reduce_to(&source);
+    let target = reduction.target_problem();
+    let best = BruteForce::new()
+        .find_best(target)
+        .expect("Knapsack target should always have an optimal solution");
+
+    assert_eq!(target.evaluate(&best), SolutionSize::Valid(5));
+
+    let extracted = reduction.extract_solution(&best);
+    assert!(!source.evaluate(&extracted));
+}
+
+#[test]
+#[should_panic(
+    expected = "Partition -> Knapsack requires all sizes and total_sum / 2 to fit in i64"
+)]
+fn test_partition_to_knapsack_panics_on_large_coefficients() {
+    let source = Partition::new(vec![(i64::MAX as u64) + 1]);
+    let _ = ReduceTo::<Knapsack>::reduce_to(&source);
+}