Fix #639: [Rule] Knapsack to ILP (#688)

* Add plan for #639: [Rule] Knapsack to ILP

* Implement #639: [Rule] Knapsack to ILP

* Add extra tests for #639 Knapsack -> ILP

* chore: remove plan file after implementation

* Fix canonical example to use rule_example_with_witness API

The direct_ilp_example helper was removed on main. Update to use the
current rule_example_with_witness API with an explicit SolutionPair.

---------

Co-authored-by: Xiwei Pan <xiwei.pan@connect.hkust-gz.edu.cn>

Author: GiggleLiu

docs/paper/reductions.typ

@@ -4676,6 +4676,42 @@ The following reductions to Integer Linear Programming are straightforward formu
   _Solution extraction._ $K = {v : x_v = 1}$.
 ]
 
+#let ks_ilp = load-example("Knapsack", "ILP")
+#let ks_ilp_sol = ks_ilp.solutions.at(0)
+#let ks_ilp_selected = ks_ilp_sol.source_config.enumerate().filter(((i, x)) => x == 1).map(((i, x)) => i)
+#let ks_ilp_sel_weight = ks_ilp_selected.fold(0, (a, i) => a + ks_ilp.source.instance.weights.at(i))
+#let ks_ilp_sel_value = ks_ilp_selected.fold(0, (a, i) => a + ks_ilp.source.instance.values.at(i))
+#reduction-rule("Knapsack", "ILP",
+  example: true,
+  example-caption: [$n = #ks_ilp.source.instance.weights.len()$ items, capacity $C = #ks_ilp.source.instance.capacity$],
+  extra: [
+    *Step 1 -- Source instance.* The canonical knapsack instance has weights $(#ks_ilp.source.instance.weights.map(str).join(", "))$, values $(#ks_ilp.source.instance.values.map(str).join(", "))$, and capacity $C = #ks_ilp.source.instance.capacity$.
+
+    *Step 2 -- Build the binary ILP.* Introduce one binary variable per item:
+    $#range(ks_ilp.source.instance.weights.len()).map(i => $x_#i$).join(", ") in {0,1}$.
+    The objective is
+    $ max #ks_ilp.source.instance.values.enumerate().map(((i, v)) => $#v x_#i$).join($+$) $
+    subject to the single capacity inequality
+    $ #ks_ilp.source.instance.weights.enumerate().map(((i, w)) => $#w x_#i$).join($+$) <= #ks_ilp.source.instance.capacity $.
+
+    *Step 3 -- Verify a solution.* The ILP optimum $bold(x)^* = (#ks_ilp_sol.target_config.map(str).join(", "))$ extracts directly to the knapsack selection $bold(x)^* = (#ks_ilp_sol.source_config.map(str).join(", "))$, choosing items $\{#ks_ilp_selected.map(str).join(", ")\}$. Their total weight is $#ks_ilp_selected.map(i => str(ks_ilp.source.instance.weights.at(i))).join(" + ") = #ks_ilp_sel_weight$ and their total value is $#ks_ilp_selected.map(i => str(ks_ilp.source.instance.values.at(i))).join(" + ") = #ks_ilp_sel_value$ #sym.checkmark.
+
+    *Uniqueness:* The fixture stores one canonical optimal witness. For this instance the optimum is unique: items $\{#ks_ilp_selected.map(str).join(", ")\}$ are the only feasible choice achieving value #ks_ilp_sel_value.
+  ],
+)[
+  A 0-1 Knapsack instance is already a binary Integer Linear Program @papadimitriou-steiglitz1982: each item-selection bit becomes a binary variable, the capacity condition is a single linear inequality, and the value objective is linear. The reduction preserves the number of decision variables exactly, producing an ILP with $n$ variables and one constraint.
+][
+  _Construction._ Given nonnegative weights $w_0, dots, w_(n-1)$, nonnegative values $v_0, dots, v_(n-1)$, and capacity $C$, introduce binary variables $x_0, dots, x_(n-1) in {0,1}$ where $x_i = 1$ iff item $i$ is selected. Construct the binary ILP:
+  $ max sum_(i=0)^(n-1) v_i x_i $
+  subject to
+  $ sum_(i=0)^(n-1) w_i x_i <= C $
+  and $x_i in {0,1}$ for all $i$. The target therefore has exactly $n$ variables and one linear constraint.
+
+  _Correctness._ ($arrow.r.double$) Any feasible knapsack solution $bold(x)$ satisfies $sum_i w_i x_i <= C$, so the same binary vector is feasible for the ILP and attains identical objective value $sum_i v_i x_i$. ($arrow.l.double$) Any feasible binary ILP solution selects exactly the items with $x_i = 1$; the single inequality guarantees the chosen set fits in the knapsack, and the ILP objective equals the knapsack value. Therefore optimal solutions correspond one-to-one and preserve the optimum value.
+
+  _Solution extraction._ Identity: return the binary variable vector $bold(x)$ as the knapsack selection.
+]
+
 #reduction-rule("MaximumClique", "MaximumIndependentSet",
   example: true,
   example-caption: [Path graph $P_4$: clique in $G$ maps to independent set in complement $overline(G)$.],

docs/paper/references.bib

@@ -934,6 +934,14 @@ @article{papadimitriou1982
   doi     = {10.1145/322307.322309}
 }
 
+@book{papadimitriou-steiglitz1982,
+  author    = {Christos H. Papadimitriou and Kenneth Steiglitz},
+  title     = {Combinatorial Optimization: Algorithms and Complexity},
+  publisher = {Prentice-Hall},
+  address   = {Englewood Cliffs, NJ},
+  year      = {1982}
+}
+
 @techreport{Heidari2022,
   author      = {Heidari, Shahrokh and Dinneen, Michael J. and Delmas, Patrice},
   title       = {An Equivalent {QUBO} Model to the Minimum Multi-Way Cut Problem},

src/rules/knapsack_ilp.rs

@@ -0,0 +1,83 @@
+//! Reduction from Knapsack to ILP (Integer Linear Programming).
+//!
+//! The standard 0-1 knapsack formulation is already a binary ILP:
+//! - Variables: one binary variable per item
+//! - Constraint: the total selected weight must not exceed capacity
+//! - Objective: maximize the total selected value
+
+use crate::models::algebraic::{ILP, LinearConstraint, ObjectiveSense};
+use crate::models::misc::Knapsack;
+use crate::reduction;
+use crate::rules::traits::{ReduceTo, ReductionResult};
+
+/// Result of reducing Knapsack to ILP.
+#[derive(Debug, Clone)]
+pub struct ReductionKnapsackToILP {
+    target: ILP<bool>,
+}
+
+impl ReductionResult for ReductionKnapsackToILP {
+    type Source = Knapsack;
+    type Target = ILP<bool>;
+
+    fn target_problem(&self) -> &ILP<bool> {
+        &self.target
+    }
+
+    fn extract_solution(&self, target_solution: &[usize]) -> Vec<usize> {
+        target_solution.to_vec()
+    }
+}
+
+#[reduction(
+    overhead = {
+        num_vars = "num_items",
+        num_constraints = "1",
+    }
+)]
+impl ReduceTo<ILP<bool>> for Knapsack {
+    type Result = ReductionKnapsackToILP;
+
+    fn reduce_to(&self) -> Self::Result {
+        let num_vars = self.num_items();
+        let constraints = vec![LinearConstraint::le(
+            self.weights()
+                .iter()
+                .enumerate()
+                .map(|(i, &weight)| (i, weight as f64))
+                .collect(),
+            self.capacity() as f64,
+        )];
+        let objective = self
+            .values()
+            .iter()
+            .enumerate()
+            .map(|(i, &value)| (i, value as f64))
+            .collect();
+        let target = ILP::new(num_vars, constraints, objective, ObjectiveSense::Maximize);
+
+        ReductionKnapsackToILP { target }
+    }
+}
+
+#[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: "knapsack_to_ilp",
+        build: || {
+            crate::example_db::specs::rule_example_with_witness::<_, ILP<bool>>(
+                Knapsack::new(vec![1, 3, 4, 5], vec![1, 4, 5, 7], 7),
+                SolutionPair {
+                    source_config: vec![0, 1, 1, 0],
+                    target_config: vec![0, 1, 1, 0],
+                },
+            )
+        },
+    }]
+}
+
+#[cfg(test)]
+#[path = "../unit_tests/rules/knapsack_ilp.rs"]
+mod tests;

src/rules/mod.rs

@@ -55,6 +55,8 @@ mod ilp_bool_ilp_i32;
 #[cfg(feature = "ilp-solver")]
 pub(crate) mod ilp_qubo;
 #[cfg(feature = "ilp-solver")]
+pub(crate) mod knapsack_ilp;
+#[cfg(feature = "ilp-solver")]
 pub(crate) mod longestcommonsubsequence_ilp;
 #[cfg(feature = "ilp-solver")]
 pub(crate) mod maximumclique_ilp;
@@ -113,6 +115,7 @@ pub(crate) fn canonical_rule_example_specs() -> Vec<crate::example_db::specs::Ru
         specs.extend(coloring_ilp::canonical_rule_example_specs());
         specs.extend(factoring_ilp::canonical_rule_example_specs());
         specs.extend(ilp_qubo::canonical_rule_example_specs());
+        specs.extend(knapsack_ilp::canonical_rule_example_specs());
         specs.extend(longestcommonsubsequence_ilp::canonical_rule_example_specs());
         specs.extend(maximumclique_ilp::canonical_rule_example_specs());
         specs.extend(maximummatching_ilp::canonical_rule_example_specs());

src/unit_tests/rules/analysis.rs

@@ -245,6 +245,8 @@ 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\"}"),
+        // Knapsack -> ILP -> QUBO is better than the direct penalty reduction
+        ("Knapsack", "QUBO {weight: \"f64\"}"),
         // MaxMatching → MaxSetPacking → ILP is better than direct MaxMatching → ILP
         (
             "MaximumMatching {graph: \"SimpleGraph\", weight: \"i32\"}",

src/unit_tests/rules/graph.rs

@@ -1,6 +1,7 @@
 use super::*;
-use crate::models::algebraic::QUBO;
+use crate::models::algebraic::{ILP, QUBO};
 use crate::models::graph::{MaximumIndependentSet, MinimumVertexCover};
+use crate::models::misc::Knapsack;
 use crate::models::set::MaximumSetPacking;
 use crate::rules::cost::{Minimize, MinimizeSteps};
 use crate::rules::graph::{classify_problem_category, ReductionStep};
@@ -50,6 +51,29 @@ fn test_find_shortest_path() {
     assert_eq!(path.len(), 1); // Direct path exists
 }
 
+#[test]
+fn test_knapsack_to_ilp_path_exists() {
+    let graph = ReductionGraph::new();
+    let src = ReductionGraph::variant_to_map(&Knapsack::variant());
+    let dst = ReductionGraph::variant_to_map(&ILP::<bool>::variant());
+    let path = graph.find_cheapest_path(
+        "Knapsack",
+        &src,
+        "ILP",
+        &dst,
+        &ProblemSize::new(vec![]),
+        &MinimizeSteps,
+    );
+
+    let path = path.expect("Knapsack should reduce to ILP");
+    assert_eq!(
+        path.type_names(),
+        vec!["Knapsack", "ILP"],
+        "Knapsack should have a direct ILP reduction"
+    );
+    assert_eq!(path.len(), 1, "Knapsack -> ILP should be one direct step");
+}
+
 #[test]
 fn test_has_direct_reduction() {
     let graph = ReductionGraph::new();

src/unit_tests/rules/knapsack_ilp.rs

@@ -0,0 +1,97 @@
+use super::*;
+use crate::models::algebraic::{Comparison, ObjectiveSense, ILP};
+use crate::rules::test_helpers::assert_optimization_round_trip_from_optimization_target;
+use crate::solvers::ILPSolver;
+
+#[test]
+fn test_knapsack_to_ilp_closed_loop() {
+    let knapsack = Knapsack::new(vec![1, 3, 4, 5], vec![1, 4, 5, 7], 7);
+    let reduction = ReduceTo::<ILP<bool>>::reduce_to(&knapsack);
+
+    assert_optimization_round_trip_from_optimization_target(
+        &knapsack,
+        &reduction,
+        "Knapsack->ILP closed loop",
+    );
+
+    let ilp_solution = ILPSolver::new()
+        .solve(reduction.target_problem())
+        .expect("ILP should be solvable");
+    let extracted = reduction.extract_solution(&ilp_solution);
+    assert_eq!(extracted, vec![0, 1, 1, 0]);
+}
+
+#[test]
+fn test_knapsack_to_ilp_structure() {
+    let knapsack = Knapsack::new(vec![1, 3, 4, 5], vec![1, 4, 5, 7], 7);
+    let reduction = ReduceTo::<ILP<bool>>::reduce_to(&knapsack);
+    let ilp = reduction.target_problem();
+
+    assert_eq!(ilp.num_vars(), 4);
+    assert_eq!(ilp.num_constraints(), 1);
+    assert_eq!(ilp.sense, ObjectiveSense::Maximize);
+    assert_eq!(ilp.objective, vec![(0, 1.0), (1, 4.0), (2, 5.0), (3, 7.0)]);
+
+    let constraint = &ilp.constraints[0];
+    assert_eq!(constraint.cmp, Comparison::Le);
+    assert_eq!(constraint.rhs, 7.0);
+    assert_eq!(
+        constraint.terms,
+        vec![(0, 1.0), (1, 3.0), (2, 4.0), (3, 5.0)]
+    );
+}
+
+#[test]
+fn test_knapsack_to_ilp_zero_capacity() {
+    let knapsack = Knapsack::new(vec![2, 3], vec![5, 7], 0);
+    let reduction = ReduceTo::<ILP<bool>>::reduce_to(&knapsack);
+
+    let ilp_solution = ILPSolver::new()
+        .solve(reduction.target_problem())
+        .expect("zero-capacity ILP should still be solvable");
+    let extracted = reduction.extract_solution(&ilp_solution);
+    assert_eq!(extracted, vec![0, 0]);
+}
+
+#[test]
+fn test_knapsack_to_ilp_empty_instance() {
+    let knapsack = Knapsack::new(vec![], vec![], 0);
+    let reduction = ReduceTo::<ILP<bool>>::reduce_to(&knapsack);
+    let ilp = reduction.target_problem();
+
+    assert_eq!(ilp.num_vars(), 0);
+    assert_eq!(ilp.num_constraints(), 1);
+    assert_eq!(ilp.constraints[0].cmp, Comparison::Le);
+    assert_eq!(ilp.constraints[0].rhs, 0.0);
+    assert!(ilp.constraints[0].terms.is_empty());
+    assert!(ilp.objective.is_empty());
+
+    let ilp_solution = ILPSolver::new()
+        .solve(ilp)
+        .expect("empty Knapsack ILP should still be solvable");
+    let extracted = reduction.extract_solution(&ilp_solution);
+    assert_eq!(extracted, Vec::<usize>::new());
+}
+
+#[cfg(feature = "example-db")]
+#[test]
+fn test_knapsack_to_ilp_canonical_example_spec() {
+    let spec = canonical_rule_example_specs()
+        .into_iter()
+        .find(|spec| spec.id == "knapsack_to_ilp")
+        .expect("missing canonical Knapsack -> ILP example spec");
+    let example = (spec.build)();
+
+    assert_eq!(example.source.problem, "Knapsack");
+    assert_eq!(example.target.problem, "ILP");
+    assert_eq!(example.source.instance["capacity"], 7);
+    assert_eq!(example.target.instance["num_vars"], 4);
+    assert_eq!(example.target.instance["constraints"].as_array().unwrap().len(), 1);
+    assert_eq!(
+        example.solutions,
+        vec![crate::export::SolutionPair {
+            source_config: vec![0, 1, 1, 0],
+            target_config: vec![0, 1, 1, 0],
+        }]
+    );
+}