From 5707218190a714ac460e5301d5e78f41e7320dc9 Mon Sep 17 00:00:00 2001
From: Xiwei Pan <xiwei.pan@connect.hkust-gz.edu.cn>
Date: Fri, 18 Sep 2026 00:34:14 +0800
Subject: [PATCH] Fix set splitting reduction for repeated elements

---
 src/models/set/set_splitting.rs               |  5 ++-
 src/rules/setsplitting_betweenness.rs         | 13 ++++++--
 .../rules/setsplitting_betweenness.rs         | 31 +++++++++++++++++++
 3 files changed, 46 insertions(+), 3 deletions(-)

diff --git a/src/models/set/set_splitting.rs b/src/models/set/set_splitting.rs
index 1c7f46a76..eaf326f36 100644
--- a/src/models/set/set_splitting.rs
+++ b/src/models/set/set_splitting.rs
@@ -68,6 +68,9 @@ fn normalize_subsets(universe_size: usize, subsets: &[Vec<usize>]) -> (usize, Ve
 
     for subset in subsets {
         let mut remainder = subset.clone();
+        // Repeated occurrences represent the same set element.
+        remainder.sort_unstable();
+        remainder.dedup();
         while remainder.len() > 3 {
             let positive_aux = next_element;
             let negative_aux = next_element + 1;
@@ -154,7 +157,7 @@ impl SetSplitting {
         (universe_size, size2, size3)
     }
 
-    /// Universe size after decomposing all subsets to size 2 or 3.
+    /// Universe size after deduplicating subsets and decomposing sizes above 3.
     pub fn normalized_universe_size(&self) -> usize {
         self.normalized_stats().0
     }
diff --git a/src/rules/setsplitting_betweenness.rs b/src/rules/setsplitting_betweenness.rs
index a62f799d4..7e6ec7963 100644
--- a/src/rules/setsplitting_betweenness.rs
+++ b/src/rules/setsplitting_betweenness.rs
@@ -1,11 +1,12 @@
 //! Reduction from Set Splitting to Betweenness.
 //!
-//! Decompose each subset to size 2 or 3 using complementarity pairs, then
+//! Deduplicate each subset and decompose sizes above 3 using complementarity pairs, then
 //! place a single pole element `p` in the Betweenness instance. A size-2
 //! subset `{u, v}` becomes `(u, p, v)`, forcing opposite sides of the pole.
 //! A size-3 subset `{u, v, w}` becomes `(u, d, v)` and `(d, p, w)` with one
 //! fresh auxiliary element `d`, which is satisfiable exactly when the three
 //! elements are not monochromatic with respect to the pole.
+//! A singleton becomes two incompatible order constraints, preserving infeasibility.
 
 use crate::models::misc::Betweenness;
 use crate::models::set::SetSplitting;
@@ -63,6 +64,14 @@ impl ReduceTo<Betweenness> for SetSplitting {
 
         for subset in normalized_subsets {
             match subset.as_slice() {
+                [u] => {
+                    // A singleton cannot contain both colors. These orders
+                    // cannot both hold for three distinct elements.
+                    let auxiliary = num_elements;
+                    num_elements += 1;
+                    triples.push((*u, pole, auxiliary));
+                    triples.push((pole, *u, auxiliary));
+                }
                 [u, v] => triples.push((*u, pole, *v)),
                 [u, v, w] => {
                     let auxiliary = num_elements;
@@ -75,7 +84,7 @@ impl ReduceTo<Betweenness> for SetSplitting {
                         SetSplitting,
                         Betweenness,
                     >(
-                        "normalized subset must contain two or three elements"
+                        "normalized subset must contain one, two or three elements",
                     ));
                 }
             }
diff --git a/src/unit_tests/rules/setsplitting_betweenness.rs b/src/unit_tests/rules/setsplitting_betweenness.rs
index b43fa7fb1..75a56fb01 100644
--- a/src/unit_tests/rules/setsplitting_betweenness.rs
+++ b/src/unit_tests/rules/setsplitting_betweenness.rs
@@ -31,6 +31,37 @@ fn test_setsplitting_to_betweenness_closed_loop() {
     );
 }
 
+#[test]
+fn test_repeated_elements_preserve_splittability() {
+    for (subset, feasible) in [
+        (vec![0, 0, 1], true),
+        (vec![1, 0, 1, 0, 1], true),
+        (vec![0, 0], false),
+        (vec![1, 1, 1, 1], false),
+    ] {
+        let source = SetSplitting::new(2, vec![subset]);
+        assert_eq!(
+            BruteForce::new().solve(&source).unwrap().is_some(),
+            feasible
+        );
+        let reduction = ReduceTo::<Betweenness>::reduce_to(&source).unwrap();
+        assert_eq!(
+            BruteForce::new()
+                .solve(reduction.target_problem())
+                .unwrap()
+                .is_some(),
+            feasible
+        );
+        if feasible {
+            assert_satisfaction_round_trip_from_satisfaction_target(
+                &source,
+                &reduction,
+                "repeated elements",
+            );
+        }
+    }
+}
+
 #[test]
 fn test_setsplitting_to_betweenness_issue_yes_instance_structure() {
     let source = issue_yes_instance();