Fix #210: [Model] Partition

docs/paper/reductions.typ

@@ -111,6 +111,7 @@
   "LongestCommonSubsequence": [Longest Common Subsequence],
   "ExactCoverBy3Sets": [Exact Cover by 3-Sets],
   "SubsetSum": [Subset Sum],
+  "Partition": [Partition],
   "MinimumFeedbackArcSet": [Minimum Feedback Arc Set],
   "MinimumFeedbackVertexSet": [Minimum Feedback Vertex Set],
   "MinimumCutIntoBoundedSets": [Minimum Cut Into Bounded Sets],
@@ -2933,6 +2934,14 @@ A classical NP-complete problem from Garey and Johnson @garey1979[Ch.~3, p.~76],
   ]
 }
 
+#problem-def("Partition")[
+  Given a finite set $A = {a_0, dots, a_(n-1)}$ with sizes $s(a_i) in ZZ^+$, determine whether there exists a subset $A' subset.eq A$ such that $sum_(a in A') s(a) = sum_(a in A without A') s(a)$.
+][
+  One of Karp's 21 NP-complete problems @karp1972, listed as SP12 in Garey & Johnson @garey1979. Partition is the special case of Subset Sum where the target equals half the total sum. Though NP-complete, it is only _weakly_ NP-hard: a dynamic-programming algorithm runs in $O(n dot B_"total")$ pseudo-polynomial time, where $B_"total" = sum_i s(a_i)$. The best known exact algorithm is the $O^*(2^(n slash 2))$ meet-in-the-middle approach of Schroeppel and Shamir (1981).
+
+  *Example.* Let $A = {3, 1, 1, 2, 2, 1}$ ($n = 6$, total sum $= 10$). Setting $A' = {3, 2}$ (indices 0, 3) gives sum $3 + 2 = 5 = 10 slash 2$, and $A without A' = {1, 1, 2, 1}$ also sums to 5. Hence a balanced partition exists.
+]
+
 #{
   let x = load-model-example("ShortestCommonSupersequence")
   let alpha-size = x.instance.alphabet_size

src/lib.rs

@@ -62,7 +62,7 @@ pub mod prelude {
     pub use crate::models::misc::{
         BinPacking, CbqRelation, ConjunctiveBooleanQuery, ConjunctiveQueryFoldability, Factoring,
         FlowShopScheduling, Knapsack, LongestCommonSubsequence, MinimumTardinessSequencing,
-        MultiprocessorScheduling, PaintShop, QueryArg, RectilinearPictureCompression,
+        MultiprocessorScheduling, PaintShop, Partition, QueryArg, RectilinearPictureCompression,
         ResourceConstrainedScheduling, SequencingToMinimizeMaximumCumulativeCost,
         SequencingWithReleaseTimesAndDeadlines, SequencingWithinIntervals,
         ShortestCommonSupersequence, StaffScheduling, StringToStringCorrection, SubsetSum,

src/models/misc/mod.rs

@@ -11,6 +11,7 @@
 //! - [`LongestCommonSubsequence`]: Longest Common Subsequence
 //! - [`MinimumTardinessSequencing`]: Minimize tardy tasks in single-machine scheduling
 //! - [`PaintShop`]: Minimize color switches in paint shop scheduling
+//! - [`Partition`]: Partition a multiset into two equal-sum subsets
 //! - [`PartiallyOrderedKnapsack`]: Knapsack with precedence constraints
 //! - [`PrecedenceConstrainedScheduling`]: Schedule unit tasks on processors by deadline
 //! - [`RectilinearPictureCompression`]: Cover 1-entries with bounded rectangles
@@ -34,6 +35,7 @@ mod minimum_tardiness_sequencing;
 mod multiprocessor_scheduling;
 pub(crate) mod paintshop;
 pub(crate) mod partially_ordered_knapsack;
+pub(crate) mod partition;
 mod precedence_constrained_scheduling;
 mod rectilinear_picture_compression;
 pub(crate) mod resource_constrained_scheduling;
@@ -57,6 +59,7 @@ pub use minimum_tardiness_sequencing::MinimumTardinessSequencing;
 pub use multiprocessor_scheduling::MultiprocessorScheduling;
 pub use paintshop::PaintShop;
 pub use partially_ordered_knapsack::PartiallyOrderedKnapsack;
+pub use partition::Partition;
 pub use precedence_constrained_scheduling::PrecedenceConstrainedScheduling;
 pub use rectilinear_picture_compression::RectilinearPictureCompression;
 pub use resource_constrained_scheduling::ResourceConstrainedScheduling;
@@ -78,6 +81,7 @@ pub(crate) fn canonical_model_example_specs() -> Vec<crate::example_db::specs::M
     specs.extend(longest_common_subsequence::canonical_model_example_specs());
     specs.extend(multiprocessor_scheduling::canonical_model_example_specs());
     specs.extend(paintshop::canonical_model_example_specs());
+    specs.extend(partition::canonical_model_example_specs());
     specs.extend(rectilinear_picture_compression::canonical_model_example_specs());
     specs.extend(sequencing_within_intervals::canonical_model_example_specs());
     specs.extend(staff_scheduling::canonical_model_example_specs());

src/models/misc/partition.rs

@@ -0,0 +1,130 @@
+//! Partition problem implementation.
+//!
+//! Given a finite set of positive integers, determine whether it can be
+//! partitioned into two subsets of equal sum. One of Karp's original 21
+//! NP-complete problems (1972), Garey & Johnson SP12.
+
+use crate::registry::{FieldInfo, ProblemSchemaEntry};
+use crate::traits::{Problem, SatisfactionProblem};
+use serde::{Deserialize, Serialize};
+
+inventory::submit! {
+    ProblemSchemaEntry {
+        name: "Partition",
+        display_name: "Partition",
+        aliases: &[],
+        dimensions: &[],
+        module_path: module_path!(),
+        description: "Determine whether a multiset of positive integers can be partitioned into two subsets of equal sum",
+        fields: &[
+            FieldInfo { name: "sizes", type_name: "Vec<u64>", description: "Positive integer size for each element" },
+        ],
+    }
+}
+
+/// The Partition problem.
+///
+/// Given a finite set `A` with `n` positive integer sizes, determine whether
+/// there exists a subset `A' ⊆ A` such that `∑_{a ∈ A'} s(a) = ∑_{a ∈ A\A'} s(a)`.
+///
+/// # Representation
+///
+/// Each element has a binary variable: `x_i = 1` if element `i` is in the
+/// second subset, `0` if in the first. The problem is satisfiable iff
+/// `∑_{i: x_i=1} sizes[i] = total_sum / 2`.
+///
+/// # Example
+///
+/// ```
+/// use problemreductions::models::misc::Partition;
+/// use problemreductions::{Problem, Solver, BruteForce};
+///
+/// let problem = Partition::new(vec![3, 1, 1, 2, 2, 1]);
+/// let solver = BruteForce::new();
+/// let solution = solver.find_satisfying(&problem);
+/// assert!(solution.is_some());
+/// ```
+#[derive(Debug, Clone, Serialize, Deserialize)]
+pub struct Partition {
+    sizes: Vec<u64>,
+}
+
+impl Partition {
+    /// Create a new Partition instance.
+    ///
+    /// # Panics
+    ///
+    /// Panics if `sizes` is empty or any size is zero.
+    pub fn new(sizes: Vec<u64>) -> Self {
+        assert!(!sizes.is_empty(), "Partition requires at least one element");
+        assert!(
+            sizes.iter().all(|&s| s > 0),
+            "All sizes must be positive (> 0)"
+        );
+        Self { sizes }
+    }
+
+    /// Returns the element sizes.
+    pub fn sizes(&self) -> &[u64] {
+        &self.sizes
+    }
+
+    /// Returns the number of elements.
+    pub fn num_elements(&self) -> usize {
+        self.sizes.len()
+    }
+
+    /// Returns the total sum of all sizes.
+    pub fn total_sum(&self) -> u64 {
+        self.sizes.iter().sum()
+    }
+}
+
+impl Problem for Partition {
+    const NAME: &'static str = "Partition";
+    type Metric = bool;
+
+    fn variant() -> Vec<(&'static str, &'static str)> {
+        crate::variant_params![]
+    }
+
+    fn dims(&self) -> Vec<usize> {
+        vec![2; self.num_elements()]
+    }
+
+    fn evaluate(&self, config: &[usize]) -> bool {
+        if config.len() != self.num_elements() {
+            return false;
+        }
+        if config.iter().any(|&v| v >= 2) {
+            return false;
+        }
+        let selected_sum: u64 = config
+            .iter()
+            .enumerate()
+            .filter(|(_, &x)| x == 1)
+            .map(|(i, _)| self.sizes[i])
+            .sum();
+        selected_sum * 2 == self.total_sum()
+    }
+}
+
+impl SatisfactionProblem for Partition {}
+
+crate::declare_variants! {
+    default sat Partition => "2^(num_elements / 2)",
+}
+
+#[cfg(feature = "example-db")]
+pub(crate) fn canonical_model_example_specs() -> Vec<crate::example_db::specs::ModelExampleSpec> {
+    vec![crate::example_db::specs::ModelExampleSpec {
+        id: "partition",
+        instance: Box::new(Partition::new(vec![3, 1, 1, 2, 2, 1])),
+        optimal_config: vec![1, 0, 0, 1, 0, 0],
+        optimal_value: serde_json::json!(true),
+    }]
+}
+
+#[cfg(test)]
+#[path = "../../unit_tests/models/misc/partition.rs"]
+mod tests;

src/models/mod.rs

@@ -29,7 +29,7 @@ pub use misc::PartiallyOrderedKnapsack;
 pub use misc::{
     BinPacking, CbqRelation, ConjunctiveBooleanQuery, ConjunctiveQueryFoldability, Factoring,
     FlowShopScheduling, Knapsack, LongestCommonSubsequence, MinimumTardinessSequencing,
-    MultiprocessorScheduling, PaintShop, PrecedenceConstrainedScheduling, QueryArg,
+    MultiprocessorScheduling, PaintShop, Partition, PrecedenceConstrainedScheduling, QueryArg,
     RectilinearPictureCompression, ResourceConstrainedScheduling,
     SequencingToMinimizeMaximumCumulativeCost, SequencingWithReleaseTimesAndDeadlines,
     SequencingWithinIntervals, ShortestCommonSupersequence, StaffScheduling,

src/unit_tests/models/misc/partition.rs

@@ -0,0 +1,108 @@
+use crate::models::misc::Partition;
+use crate::solvers::{BruteForce, Solver};
+use crate::traits::Problem;
+
+#[test]
+fn test_partition_basic() {
+    let problem = Partition::new(vec![3, 1, 1, 2, 2, 1]);
+    assert_eq!(problem.num_elements(), 6);
+    assert_eq!(problem.sizes(), &[3, 1, 1, 2, 2, 1]);
+    assert_eq!(problem.total_sum(), 10);
+    assert_eq!(problem.dims(), vec![2; 6]);
+}
+
+#[test]
+fn test_partition_evaluate_satisfying() {
+    let problem = Partition::new(vec![3, 1, 1, 2, 2, 1]);
+    // A' = {3, 2} (indices 0, 3), sum = 5 = 10/2
+    assert!(problem.evaluate(&[1, 0, 0, 1, 0, 0]));
+}
+
+#[test]
+fn test_partition_evaluate_unsatisfying() {
+    let problem = Partition::new(vec![3, 1, 1, 2, 2, 1]);
+    // All in first subset, selected sum = 0 != 5
+    assert!(!problem.evaluate(&[0, 0, 0, 0, 0, 0]));
+    // All in second subset, selected sum = 10 != 5
+    assert!(!problem.evaluate(&[1, 1, 1, 1, 1, 1]));
+}
+
+#[test]
+fn test_partition_evaluate_wrong_length() {
+    let problem = Partition::new(vec![3, 1, 1, 2, 2, 1]);
+    assert!(!problem.evaluate(&[1, 0, 0]));
+    assert!(!problem.evaluate(&[1, 0, 0, 1, 0, 0, 0]));
+}
+
+#[test]
+fn test_partition_evaluate_invalid_value() {
+    let problem = Partition::new(vec![3, 1, 1, 2, 2, 1]);
+    assert!(!problem.evaluate(&[2, 0, 0, 0, 0, 0]));
+}
+
+#[test]
+fn test_partition_odd_total() {
+    // Total = 7 (odd), no equal partition possible
+    let problem = Partition::new(vec![3, 1, 2, 1]);
+    let solver = BruteForce::new();
+    assert!(solver.find_satisfying(&problem).is_none());
+}
+
+#[test]
+fn test_partition_solver() {
+    let problem = Partition::new(vec![3, 1, 1, 2, 2, 1]);
+    let solver = BruteForce::new();
+    let solution = solver.find_satisfying(&problem);
+    assert!(solution.is_some());
+    assert!(problem.evaluate(&solution.unwrap()));
+}
+
+#[test]
+fn test_partition_solver_all() {
+    let problem = Partition::new(vec![3, 1, 1, 2, 2, 1]);
+    let solver = BruteForce::new();
+    let solutions = solver.find_all_satisfying(&problem);
+    // 10 satisfying configs for {3,1,1,2,2,1} with target half-sum 5
+    assert_eq!(solutions.len(), 10);
+    for sol in &solutions {
+        assert!(problem.evaluate(sol));
+    }
+}
+
+#[test]
+fn test_partition_single_element() {
+    // Single element can never be partitioned equally
+    let problem = Partition::new(vec![5]);
+    let solver = BruteForce::new();
+    assert!(solver.find_satisfying(&problem).is_none());
+}
+
+#[test]
+fn test_partition_two_equal_elements() {
+    let problem = Partition::new(vec![4, 4]);
+    let solver = BruteForce::new();
+    let solution = solver.find_satisfying(&problem);
+    assert!(solution.is_some());
+    assert!(problem.evaluate(&solution.unwrap()));
+}
+
+#[test]
+fn test_partition_serialization() {
+    let problem = Partition::new(vec![3, 1, 1, 2, 2, 1]);
+    let json = serde_json::to_value(&problem).unwrap();
+    let restored: Partition = serde_json::from_value(json).unwrap();
+    assert_eq!(restored.sizes(), problem.sizes());
+    assert_eq!(restored.num_elements(), problem.num_elements());
+}
+
+#[test]
+#[should_panic(expected = "All sizes must be positive")]
+fn test_partition_zero_size_panics() {
+    Partition::new(vec![3, 0, 1]);
+}
+
+#[test]
+#[should_panic(expected = "at least one element")]
+fn test_partition_empty_panics() {
+    Partition::new(vec![]);
+}

src/unit_tests/trait_consistency.rs

@@ -88,6 +88,7 @@ fn test_all_problems_implement_trait_correctly() {
         "BalancedCompleteBipartiteSubgraph",
     );
     check_problem_trait(&Factoring::new(6, 2, 2), "Factoring");
+    check_problem_trait(&Partition::new(vec![3, 1, 1, 2, 2, 1]), "Partition");
     check_problem_trait(
         &QuadraticAssignment::new(vec![vec![0, 1], vec![1, 0]], vec![vec![0, 1], vec![1, 0]]),
         "QuadraticAssignment",