CodingThrust
diff --git a/‎docs/paper/reductions.typ‎
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diff --git a/‎docs/paper/references.bib‎
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diff --git a/‎problemreductions-cli/src/commands/create.rs‎
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diff --git a/‎src/models/graph/minimum_vertex_cover.rs‎
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diff --git a/‎src/models/misc/ensemble_computation.rs‎
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diff --git a/‎src/models/misc/longest_common_subsequence.rs‎
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diff --git a/‎src/models/misc/paintshop.rs‎
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diff --git a/‎src/models/mod.rs‎
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diff --git a/‎src/rules/hamiltoniancircuit_longestcircuit.rs‎
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@@ -1521,6 +1521,18 @@ @techreport{plaisted1976
   year        = {1976}
 }
 
+@article{Streif2021,
+  title   = {Beating classical heuristics for the binary paint shop problem
+             with the quantum approximate optimization algorithm},
+  author  = {Streif, Michael and Yarkoni, Sheir and Skolik, Andrea
+             and Neukart, Florian and Leib, Martin},
+  journal = {Physical Review A},
+  volume  = {104},
+  pages   = {012403},
+  year    = {2021},
+  doi     = {10.1103/PhysRevA.104.012403}
+}
+
 @techreport{storer1977,
   author      = {James A. Storer},
   title       = {NP-Completeness Results Concerning Data Compression},
 
@@ -28,11 +28,11 @@ use problemreductions::models::misc::{
     LongestCommonSubsequence, MinimumExternalMacroDataCompression, MinimumTardinessSequencing,
     MultiprocessorScheduling, PaintShop, PartiallyOrderedKnapsack, ProductionPlanning, QueryArg,
     RectilinearPictureCompression, RegisterSufficiency, ResourceConstrainedScheduling,
-    SchedulingToMinimizeWeightedCompletionTime,
-    SchedulingWithIndividualDeadlines, SequencingToMinimizeMaximumCumulativeCost,
-    SequencingToMinimizeWeightedCompletionTime, SequencingToMinimizeWeightedTardiness,
-    SequencingWithReleaseTimesAndDeadlines, SequencingWithinIntervals, ShortestCommonSupersequence,
-    StringToStringCorrection, SubsetSum, SumOfSquaresPartition, ThreePartition, TimetableDesign,
+    SchedulingToMinimizeWeightedCompletionTime, SchedulingWithIndividualDeadlines,
+    SequencingToMinimizeMaximumCumulativeCost, SequencingToMinimizeWeightedCompletionTime,
+    SequencingToMinimizeWeightedTardiness, SequencingWithReleaseTimesAndDeadlines,
+    SequencingWithinIntervals, ShortestCommonSupersequence, StringToStringCorrection, SubsetSum,
+    SumOfSquaresPartition, ThreePartition, TimetableDesign,
 };
 use problemreductions::models::BiconnectivityAugmentation;
 use problemreductions::prelude::*;
@@ -661,7 +661,7 @@ fn example_for(canonical: &str, graph_type: Option<&str>) -> &'static str {
         "SpinGlass" => "--graph 0-1,1-2 --couplings 1,1",
         "KColoring" => "--graph 0-1,1-2,2-0 --k 3",
         "HamiltonianCircuit" => "--graph 0-1,1-2,2-3,3-0",
-        "EnsembleComputation" => "--universe 4 --sets \"0,1,2;0,1,3\" --budget 4",
+        "EnsembleComputation" => "--universe 4 --sets \"0,1,2;0,1,3\"",
         "RootedTreeStorageAssignment" => "--universe 5 --sets \"0,2;1,3;0,4;2,4\" --bound 1",
         "MinMaxMulticenter" => {
             "--graph 0-1,1-2,2-3 --weights 1,1,1,1 --edge-weights 1,1,1 --k 2"
@@ -2622,26 +2622,23 @@ pub fn create(args: &CreateArgs, out: &OutputConfig) -> Result<()> {
         // EnsembleComputation
         "EnsembleComputation" => {
             let usage =
-                "Usage: pred create EnsembleComputation --universe 4 --sets \"0,1,2;0,1,3\" --budget 4";
+                "Usage: pred create EnsembleComputation --universe 4 --sets \"0,1,2;0,1,3\" [--budget 4]";
             let universe_size = args.universe.ok_or_else(|| {
                 anyhow::anyhow!("EnsembleComputation requires --universe\n\n{usage}")
             })?;
             let subsets = parse_sets(args)?;
-            let budget = args
-                .budget
-                .as_deref()
-                .ok_or_else(|| anyhow::anyhow!("EnsembleComputation requires --budget\n\n{usage}"))?
-                .parse::<usize>()
-                .map_err(|e| {
+            let instance = if let Some(budget_str) = args.budget.as_deref() {
+                let budget = budget_str.parse::<usize>().map_err(|e| {
                     anyhow::anyhow!(
                         "Invalid --budget value for EnsembleComputation: {e}\n\n{usage}"
                     )
                 })?;
-            (
-                ser(EnsembleComputation::try_new(universe_size, subsets, budget)
-                    .map_err(anyhow::Error::msg)?)?,
-                resolved_variant.clone(),
-            )
+                EnsembleComputation::try_new(universe_size, subsets, budget)
+                    .map_err(anyhow::Error::msg)?
+            } else {
+                EnsembleComputation::with_default_budget(universe_size, subsets)
+            };
+            (ser(instance)?, resolved_variant.clone())
         }
 
         // ComparativeContainment
 
@@ -6,7 +6,7 @@
 use crate::registry::{FieldInfo, ProblemSchemaEntry, VariantDimension};
 use crate::topology::{Graph, SimpleGraph};
 use crate::traits::Problem;
-use crate::types::{Min, WeightElement};
+use crate::types::{Min, One, WeightElement};
 use num_traits::Zero;
 use serde::{Deserialize, Serialize};
 
@@ -17,7 +17,7 @@ inventory::submit! {
         aliases: &["MVC"],
         dimensions: &[
             VariantDimension::new("graph", "SimpleGraph", &["SimpleGraph"]),
-            VariantDimension::new("weight", "i32", &["i32"]),
+            VariantDimension::new("weight", "i32", &["i32", "One"]),
         ],
         module_path: module_path!(),
         description: "Find minimum weight vertex cover in a graph",
@@ -152,6 +152,7 @@ fn is_vertex_cover_config<G: Graph>(graph: &G, config: &[usize]) -> bool {
 
 crate::declare_variants! {
     default MinimumVertexCover<SimpleGraph, i32> => "1.1996^num_vertices",
+    MinimumVertexCover<SimpleGraph, One> => "1.1996^num_vertices",
 }
 
 #[cfg(feature = "example-db")]
 
@@ -2,6 +2,7 @@
 
 use crate::registry::{FieldInfo, ProblemSchemaEntry};
 use crate::traits::Problem;
+use crate::types::Min;
 use serde::{Deserialize, Serialize};
 
 inventory::submit! {
@@ -11,7 +12,7 @@ inventory::submit! {
         aliases: &[],
         dimensions: &[],
         module_path: module_path!(),
-        description: "Determine whether required subsets can be built by a bounded sequence of disjoint unions",
+        description: "Find the minimum-length sequence of disjoint unions that builds all required subsets",
         fields: &[
             FieldInfo { name: "universe_size", type_name: "usize", description: "Number of elements in the universe A" },
             FieldInfo { name: "subsets", type_name: "Vec<Vec<usize>>", description: "Required subsets that must appear among the computed z_i values" },
@@ -33,6 +34,23 @@ impl EnsembleComputation {
         Self::try_new(universe_size, subsets, budget).unwrap_or_else(|err| panic!("{err}"))
     }
 
+    /// Create with an automatically derived search-space bound.
+    ///
+    /// The default budget is the sum of all subset sizes (worst-case without
+    /// intermediate-set reuse). This is always sufficient for the optimal
+    /// solution to fit within the search space.
+    pub fn with_default_budget(universe_size: usize, subsets: Vec<Vec<usize>>) -> Self {
+        let budget = Self::default_budget(&subsets);
+        Self::new(universe_size, subsets, budget)
+    }
+
+    /// Compute a default search-space bound from the subsets.
+    ///
+    /// Returns the sum of all subset sizes, clamped to at least 1.
+    pub fn default_budget(subsets: &[Vec<usize>]) -> usize {
+        subsets.iter().map(|s| s.len()).sum::<usize>().max(1)
+    }
+
     pub fn try_new(
         universe_size: usize,
         subsets: Vec<Vec<usize>>,
@@ -146,49 +164,47 @@ impl EnsembleComputation {
 
 impl Problem for EnsembleComputation {
     const NAME: &'static str = "EnsembleComputation";
-    type Value = crate::types::Or;
+    type Value = Min<usize>;
 
     fn dims(&self) -> Vec<usize> {
         vec![self.universe_size + self.budget; 2 * self.budget]
     }
 
-    fn evaluate(&self, config: &[usize]) -> crate::types::Or {
-        crate::types::Or({
-            if config.len() != 2 * self.budget {
-                return crate::types::Or(false);
-            }
+    fn evaluate(&self, config: &[usize]) -> Min<usize> {
+        if config.len() != 2 * self.budget {
+            return Min(None);
+        }
 
-            let Some(required_subsets) = self.required_subsets() else {
-                return crate::types::Or(false);
+        let Some(required_subsets) = self.required_subsets() else {
+            return Min(None);
+        };
+        if required_subsets.is_empty() {
+            return Min(Some(0));
+        }
+
+        let mut computed = Vec::with_capacity(self.budget);
+        for step in 0..self.budget {
+            let left_operand = config[2 * step];
+            let right_operand = config[2 * step + 1];
+
+            let Some(left) = self.decode_operand(left_operand, &computed) else {
+                return Min(None);
+            };
+            let Some(right) = self.decode_operand(right_operand, &computed) else {
+                return Min(None);
             };
-            if required_subsets.is_empty() {
-                return crate::types::Or(true);
+
+            if !Self::are_disjoint(&left, &right) {
+                return Min(None);
             }
 
-            let mut computed = Vec::with_capacity(self.budget);
-            for step in 0..self.budget {
-                let left_operand = config[2 * step];
-                let right_operand = config[2 * step + 1];
-
-                let Some(left) = self.decode_operand(left_operand, &computed) else {
-                    return crate::types::Or(false);
-                };
-                let Some(right) = self.decode_operand(right_operand, &computed) else {
-                    return crate::types::Or(false);
-                };
-
-                if !Self::are_disjoint(&left, &right) {
-                    return crate::types::Or(false);
-                }
-
-                computed.push(Self::union_disjoint(&left, &right));
-                if Self::all_required_subsets_present(&required_subsets, &computed) {
-                    return crate::types::Or(true);
-                }
+            computed.push(Self::union_disjoint(&left, &right));
+            if Self::all_required_subsets_present(&required_subsets, &computed) {
+                return Min(Some(step + 1));
             }
+        }
 
-            false
-        })
+        Min(None)
     }
 
     fn variant() -> Vec<(&'static str, &'static str)> {
@@ -227,7 +243,7 @@ pub(crate) fn canonical_model_example_specs() -> Vec<crate::example_db::specs::M
             2,
         )),
         optimal_config: vec![0, 1, 3, 2],
-        optimal_value: serde_json::json!(true),
+        optimal_value: serde_json::json!(2),
     }]
 }
 
 
@@ -118,6 +118,25 @@ impl LongestCommonSubsequence {
     pub fn num_transitions(&self) -> usize {
         self.max_length.saturating_sub(1)
     }
+
+    /// Returns the cross-frequency product: the sum over each alphabet symbol
+    /// of the product of that symbol's frequency across all input strings.
+    ///
+    /// Formally: Σ_{c ∈ 0..alphabet_size} Π_{i=1..k} count(c, strings\[i\])
+    /// where count(c, s) is the number of occurrences of symbol c in string s.
+    ///
+    /// This equals the exact number of match-node vertices in the LCS → MaxIS
+    /// reduction graph.
+    pub fn cross_frequency_product(&self) -> usize {
+        (0..self.alphabet_size)
+            .map(|c| {
+                self.strings
+                    .iter()
+                    .map(|s| s.iter().filter(|&&sym| sym == c).count())
+                    .product::<usize>()
+            })
+            .sum()
+    }
 }
 
 /// Check whether `candidate` is a subsequence of `target` using greedy
 
@@ -134,6 +134,16 @@ impl PaintShop {
         &self.car_labels
     }
 
+    /// Get the sequence as car indices.
+    pub fn sequence_indices(&self) -> &[usize] {
+        &self.sequence_indices
+    }
+
+    /// Get whether each position is the first occurrence of its car.
+    pub fn is_first(&self) -> &[bool] {
+        &self.is_first
+    }
+
     /// Get the coloring of the sequence from a configuration.
     ///
     /// Config assigns a color (0 or 1) to each car for its first occurrence.
 
@@ -45,12 +45,11 @@ pub use misc::{
     MultiprocessorScheduling, PaintShop, Partition, PrecedenceConstrainedScheduling,
     ProductionPlanning, QueryArg, RectilinearPictureCompression, RegisterSufficiency,
     ResourceConstrainedScheduling, SchedulingToMinimizeWeightedCompletionTime,
-    SchedulingWithIndividualDeadlines,
-    SequencingToMinimizeMaximumCumulativeCost, SequencingToMinimizeWeightedCompletionTime,
-    SequencingToMinimizeWeightedTardiness, SequencingWithReleaseTimesAndDeadlines,
-    SequencingWithinIntervals, ShortestCommonSupersequence, StackerCrane, StaffScheduling,
-    StringToStringCorrection, SubsetSum, SumOfSquaresPartition, Term, ThreePartition,
-    TimetableDesign,
+    SchedulingWithIndividualDeadlines, SequencingToMinimizeMaximumCumulativeCost,
+    SequencingToMinimizeWeightedCompletionTime, SequencingToMinimizeWeightedTardiness,
+    SequencingWithReleaseTimesAndDeadlines, SequencingWithinIntervals, ShortestCommonSupersequence,
+    StackerCrane, StaffScheduling, StringToStringCorrection, SubsetSum, SumOfSquaresPartition,
+    Term, ThreePartition, TimetableDesign,
 };
 pub use set::{
     ComparativeContainment, ConsecutiveSets, ExactCoverBy3Sets, IntegerKnapsack, MaximumSetPacking,
 
@@ -0,0 +1,69 @@
+//! Reduction from HamiltonianCircuit to LongestCircuit.
+//!
+//! Given an HC instance G = (V, E), construct an LC instance on the same graph
+//! with unit edge weights. A Hamiltonian circuit exists iff the optimal circuit
+//! length equals |V|.
+
+use crate::models::graph::{HamiltonianCircuit, LongestCircuit};
+use crate::reduction;
+use crate::rules::traits::{ReduceTo, ReductionResult};
+use crate::topology::{Graph, SimpleGraph};
+
+/// Result of reducing HamiltonianCircuit to LongestCircuit.
+#[derive(Debug, Clone)]
+pub struct ReductionHamiltonianCircuitToLongestCircuit {
+    target: LongestCircuit<SimpleGraph, i32>,
+}
+
+impl ReductionResult for ReductionHamiltonianCircuitToLongestCircuit {
+    type Source = HamiltonianCircuit<SimpleGraph>;
+    type Target = LongestCircuit<SimpleGraph, i32>;
+
+    fn target_problem(&self) -> &Self::Target {
+        &self.target
+    }
+
+    fn extract_solution(&self, target_solution: &[usize]) -> Vec<usize> {
+        crate::rules::graph_helpers::edges_to_cycle_order(self.target.graph(), target_solution)
+    }
+}
+
+#[reduction(
+    overhead = {
+        num_vertices = "num_vertices",
+        num_edges = "num_edges",
+    }
+)]
+impl ReduceTo<LongestCircuit<SimpleGraph, i32>> for HamiltonianCircuit<SimpleGraph> {
+    type Result = ReductionHamiltonianCircuitToLongestCircuit;
+
+    fn reduce_to(&self) -> Self::Result {
+        let n = self.num_vertices();
+        let edges = self.graph().edges();
+        let target = LongestCircuit::new(SimpleGraph::new(n, edges), vec![1i32; self.num_edges()]);
+        ReductionHamiltonianCircuitToLongestCircuit { 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: "hamiltoniancircuit_to_longestcircuit",
+        build: || {
+            let source = HamiltonianCircuit::new(SimpleGraph::cycle(4));
+            crate::example_db::specs::rule_example_with_witness::<_, LongestCircuit<SimpleGraph, i32>>(
+                source,
+                SolutionPair {
+                    source_config: vec![0, 1, 2, 3],
+                    target_config: vec![1, 1, 1, 1],
+                },
+            )
+        },
+    }]
+}
+
+#[cfg(test)]
+#[path = "../unit_tests/rules/hamiltoniancircuit_longestcircuit.rs"]
+mod tests;