Fix #444: [Model] MinimumCardinalityKey (#679)

* Add plan for #444: MinimumCardinalityKey model

* Implement #444: MinimumCardinalityKey model

- Add MinimumCardinalityKey satisfaction problem (set/ category)
- Functional dependency closure under Armstrong's axioms
- CLI create support with --num-attributes, --dependencies, --k flags
- Paper entry with problem definition and worked example
- 11 unit tests covering creation, evaluation, solver, serialization
- Regenerate example-db fixtures

* chore: remove plan file after implementation

* Fix MinimumCardinalityKey review feedback

* Fix formatting after merge conflict resolution

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

* Add missing fields to empty_args() test helper after merge

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

---------

Co-authored-by: Xiwei Pan <xiwei.pan@connect.hkust-gz.edu.cn>
Co-authored-by: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

Author: 3 people

docs/paper/reductions.typ

@@ -80,6 +80,7 @@
   "MinimumSetCovering": [Minimum Set Covering],
   "ComparativeContainment": [Comparative Containment],
   "SetBasis": [Set Basis],
+  "MinimumCardinalityKey": [Minimum Cardinality Key],
   "SpinGlass": [Spin Glass],
   "QUBO": [QUBO],
   "ILP": [Integer Linear Programming],
@@ -1518,6 +1519,29 @@ NP-completeness was established by Garey, Johnson, and Stockmeyer @gareyJohnsonS
   ]
 }
 
+#{
+  let x = load-model-example("MinimumCardinalityKey")
+  let n = x.instance.num_attributes
+  let deps = x.instance.dependencies
+  let m = deps.len()
+  let bound = x.instance.bound_k
+  let sample = x.samples.at(0)
+  let key-attrs = range(n).filter(i => sample.config.at(i) == 1)
+  let sat-count = x.optimal.len()
+  let fmt-set(s) = "${" + s.map(e => str(e)).join(", ") + "}$"
+  let fmt-fd(d) = fmt-set(d.at(0)) + " $arrow.r$ " + fmt-set(d.at(1))
+  [
+    #problem-def("MinimumCardinalityKey")[
+      Given a set $A$ of attribute names, a collection $F$ of functional dependencies (ordered pairs of subsets of $A$), and a positive integer $M$, does there exist a candidate key $K subset.eq A$ with $|K| <= M$, i.e., a minimal subset $K$ such that the closure of $K$ under $F^*$ equals $A$?
+    ][
+    The Minimum Cardinality Key problem arises in relational database theory, where identifying the smallest candidate key determines the most efficient way to uniquely identify rows in a relation. It was shown NP-complete by Lucchesi and Osborn (1978) @lucchesi1978keys via transformation from Vertex Cover. The problem appears as SR26 in Garey & Johnson (A4) @garey1979. The closure $F^*$ is defined by Armstrong's axioms: reflexivity ($B subset.eq C$ implies $C arrow.r B$), transitivity, and union. The best known exact algorithm is brute-force enumeration of all subsets of $A$, giving $O^*(2^(|A|))$ time#footnote[Lucchesi and Osborn give an output-polynomial algorithm for enumerating all candidate keys, but the number of keys can be exponential.].
+
+    *Example.* Let $A = {0, 1, ..., #(n - 1)}$ ($|A| = #n$) with $M = #bound$ and functional dependencies $F = {#deps.enumerate().map(((i, d)) => fmt-fd(d)).join(", ")}$.
+    The candidate key $K = #fmt-set(key-attrs)$ has $|K| = #key-attrs.len() <= #bound$. Its closure: start with ${0, 1}$; apply ${0, 1} arrow.r {2}$ to get ${0, 1, 2}$; apply ${0, 2} arrow.r {3}$ to get ${0, 1, 2, 3}$; apply ${1, 3} arrow.r {4}$ to get ${0, 1, 2, 3, 4}$; apply ${2, 4} arrow.r {5}$ to get $A$. Neither ${0}$ nor ${1}$ alone determines $A$, so $K$ is minimal. There are #sat-count satisfying encodings in total.
+    ]
+  ]
+}
+
 == Optimization Problems
 
 #{

docs/paper/references.bib

@@ -710,6 +710,17 @@ @article{lucchesi1978
   doi     = {10.1112/jlms/s2-17.3.369}
 }
 
+@article{lucchesi1978keys,
+  author  = {Cl\'audio L. Lucchesi and Sylvia L. Osborn},
+  title   = {Candidate Keys for Relations},
+  journal = {Journal of Computer and System Sciences},
+  volume  = {17},
+  number  = {2},
+  pages   = {270--279},
+  year    = {1978},
+  doi     = {10.1016/0022-0000(78)90009-0}
+}
+
 @article{lenstra1976,
   author  = {Jan Karel Lenstra and Alexander H. G. Rinnooy Kan},
   title   = {On General Routing Problems},

docs/src/cli.md

@@ -352,6 +352,7 @@ pred create LengthBoundedDisjointPaths --graph 0-1,1-6,0-2,2-3,3-6,0-4,4-5,5-6 -
 pred create Factoring --target 15 --bits-m 4 --bits-n 4 -o factoring.json
 pred create Factoring --target 21 --bits-m 3 --bits-n 3 -o factoring2.json
 pred create X3C --universe 9 --sets "0,1,2;0,2,4;3,4,5;3,5,7;6,7,8;1,4,6;2,5,8" -o x3c.json
+pred create MinimumCardinalityKey --num-attributes 6 --dependencies "0,1>2;0,2>3;1,3>4;2,4>5" --k 2 -o mck.json
 pred create MinimumTardinessSequencing --n 5 --deadlines 5,5,5,3,3 --precedence-pairs "0>3,1>3,1>4,2>4" -o mts.json
 pred create StringToStringCorrection --source-string "0,1,2,3,1,0" --target-string "0,1,3,2,1" --bound 2 | pred solve - --solver brute-force
 pred create StrongConnectivityAugmentation --arcs "0>1,1>2,2>0,3>4,4>3,2>3,4>5,5>3" --candidate-arcs "3>0:5,3>1:3,3>2:4,4>0:6,4>1:2,4>2:7,5>0:4,5>1:3,5>2:1,0>3:8,0>4:3,0>5:2,1>3:6,1>4:4,1>5:5,2>4:3,2>5:7,1>0:2" --bound 1 -o sca.json
@@ -519,10 +520,17 @@ Source evaluation: Valid(2)
 
 > **Note:** The ILP solver requires a reduction path from the target problem to ILP.
 > Some problems do not currently have one. Examples include BoundedComponentSpanningForest,
-> LengthBoundedDisjointPaths, QUBO, SpinGlass, MaxCut, CircuitSAT, and MultiprocessorScheduling.
+> LengthBoundedDisjointPaths, MinimumCardinalityKey, QUBO, SpinGlass, MaxCut, CircuitSAT, and MultiprocessorScheduling.
 > Use `pred solve <file> --solver brute-force` for these, or reduce to a problem that supports ILP first.
 > For other problems, use `pred path <PROBLEM> ILP` to check whether an ILP reduction path exists.
 
+For example, the canonical Minimum Cardinality Key instance can be created and solved with:
+
+```bash
+pred create MinimumCardinalityKey --num-attributes 6 --dependencies "0,1>2;0,2>3;1,3>4;2,4>5" --k 2 -o mck.json
+pred solve mck.json --solver brute-force
+```
+
 ## Shell Completions
 
 Enable tab completion by adding one line to your shell config:

problemreductions-cli/src/cli.rs

@@ -238,6 +238,7 @@ Flags by problem type:
   ComparativeContainment          --universe, --r-sets, --s-sets [--r-weights] [--s-weights]
   X3C (ExactCoverBy3Sets)         --universe, --sets (3 elements each)
   SetBasis                        --universe, --sets, --k
+  MinimumCardinalityKey           --num-attributes, --dependencies, --k
   BicliqueCover                   --left, --right, --biedges, --k
   BalancedCompleteBipartiteSubgraph --left, --right, --biedges, --k
   BiconnectivityAugmentation      --graph, --potential-edges, --budget [--num-vertices]
@@ -286,7 +287,8 @@ Examples:
   pred create FVS --arcs \"0>1,1>2,2>0\" --weights 1,1,1
   pred create UndirectedTwoCommodityIntegralFlow --graph 0-2,1-2,2-3 --capacities 1,1,2 --source-1 0 --sink-1 3 --source-2 1 --sink-2 3 --requirement-1 1 --requirement-2 1
   pred create X3C --universe 9 --sets \"0,1,2;0,2,4;3,4,5;3,5,7;6,7,8;1,4,6;2,5,8\"
-  pred create SetBasis --universe 4 --sets \"0,1;1,2;0,2;0,1,2\" --k 3")]
+  pred create SetBasis --universe 4 --sets \"0,1;1,2;0,2;0,1,2\" --k 3
+  pred create MinimumCardinalityKey --num-attributes 6 --dependencies \"0,1>2;0,2>3;1,3>4;2,4>5\" --k 2")]
 pub struct CreateArgs {
     /// Problem type (e.g., MIS, QUBO, SAT). Omit when using --example.
     #[arg(value_parser = crate::problem_name::ProblemNameParser)]
@@ -498,6 +500,12 @@ pub struct CreateArgs {
     /// Alphabet size for SCS or StringToStringCorrection (optional; inferred from max symbol + 1 if omitted)
     #[arg(long)]
     pub alphabet_size: Option<usize>,
+    /// Functional dependencies for MinimumCardinalityKey (semicolon-separated "lhs>rhs" pairs, e.g., "0,1>2;0,2>3")
+    #[arg(long)]
+    pub dependencies: Option<String>,
+    /// Number of attributes for MinimumCardinalityKey
+    #[arg(long)]
+    pub num_attributes: Option<usize>,
     /// Source string for StringToStringCorrection (comma-separated symbol indices, e.g., "0,1,2,3")
     #[arg(long)]
     pub source_string: Option<String>,

problemreductions-cli/src/commands/create.rs

@@ -94,6 +94,8 @@ fn all_data_flags_empty(args: &CreateArgs) -> bool {
         && args.requirements.is_none()
         && args.num_workers.is_none()
         && args.alphabet_size.is_none()
+        && args.dependencies.is_none()
+        && args.num_attributes.is_none()
         && args.source_string.is_none()
         && args.target_string.is_none()
         && args.capacities.is_none()
@@ -239,6 +241,7 @@ fn type_format_hint(type_name: &str, graph_type: Option<&str>) -> &'static str {
             Some("UnitDiskGraph") => "float positions: \"0.0,0.0;1.0,0.0\"",
             _ => "edge list: 0-1,1-2,2-3",
         },
+        "Vec<(Vec<usize>, Vec<usize>)>" => "semicolon-separated dependencies: \"0,1>2;0,2>3\"",
         "Vec<u64>" => "comma-separated integers: 4,5,3,2,6",
         "Vec<W>" => "comma-separated: 1,2,3",
         "Vec<usize>" => "comma-separated indices: 0,2,4",
@@ -329,6 +332,9 @@ fn example_for(canonical: &str, graph_type: Option<&str>) -> &'static str {
             "--universe 4 --r-sets \"0,1,2,3;0,1\" --s-sets \"0,1,2,3;2,3\" --r-weights 2,5 --s-weights 3,6"
         }
         "SetBasis" => "--universe 4 --sets \"0,1;1,2;0,2;0,1,2\" --k 3",
+        "MinimumCardinalityKey" => {
+            "--num-attributes 6 --dependencies \"0,1>2;0,2>3;1,3>4;2,4>5\" --k 2"
+        }
         "ShortestCommonSupersequence" => "--strings \"0,1,2;1,2,0\" --bound 4",
         "StringToStringCorrection" => {
             "--source-string \"0,1,2,3,1,0\" --target-string \"0,1,3,2,1\" --bound 2"
@@ -342,6 +348,7 @@ fn help_flag_name(canonical: &str, field_name: &str) -> String {
     match (canonical, field_name) {
         ("BoundedComponentSpanningForest", "max_components") => return "k".to_string(),
         ("BoundedComponentSpanningForest", "max_weight") => return "bound".to_string(),
+        ("MinimumCardinalityKey", "bound_k") => return "k".to_string(),
         ("StaffScheduling", "shifts_per_schedule") => return "k".to_string(),
         _ => {}
     }
@@ -1223,6 +1230,33 @@ pub fn create(args: &CreateArgs, out: &OutputConfig) -> Result<()> {
             )
         }
 
+        // MinimumCardinalityKey
+        "MinimumCardinalityKey" => {
+            let num_attributes = args.num_attributes.ok_or_else(|| {
+                anyhow::anyhow!(
+                    "MinimumCardinalityKey requires --num-attributes, --dependencies, and --k\n\n\
+                     Usage: pred create MinimumCardinalityKey --num-attributes 6 --dependencies \"0,1>2;0,2>3;1,3>4;2,4>5\" --k 2"
+                )
+            })?;
+            let k = args.k.ok_or_else(|| {
+                anyhow::anyhow!("MinimumCardinalityKey requires --k (bound on key cardinality)")
+            })?;
+            let deps_str = args.dependencies.as_deref().ok_or_else(|| {
+                anyhow::anyhow!(
+                    "MinimumCardinalityKey requires --dependencies (e.g., \"0,1>2;0,2>3\")"
+                )
+            })?;
+            let dependencies = parse_dependencies(deps_str)?;
+            (
+                ser(problemreductions::models::set::MinimumCardinalityKey::new(
+                    num_attributes,
+                    dependencies,
+                    k,
+                ))?,
+                resolved_variant.clone(),
+            )
+        }
+
         // BicliqueCover
         "BicliqueCover" => {
             let usage = "pred create BicliqueCover --left 2 --right 2 --biedges 0-0,0-1,1-1 --k 2";
@@ -2328,6 +2362,39 @@ fn parse_named_sets(sets_str: Option<&str>, flag: &str) -> Result<Vec<Vec<usize>
         .collect()
 }
 
+/// Parse `--dependencies` as semicolon-separated "lhs>rhs" pairs.
+/// E.g., "0,1>2;0,2>3;1,3>4;2,4>5" means {0,1}->{2}, {0,2}->{3}, etc.
+fn parse_dependencies(input: &str) -> Result<Vec<(Vec<usize>, Vec<usize>)>> {
+    fn parse_dependency_side(side: &str) -> Result<Vec<usize>> {
+        if side.trim().is_empty() {
+            return Ok(vec![]);
+        }
+        side.split(',')
+            .map(|s| {
+                s.trim()
+                    .parse::<usize>()
+                    .map_err(|e| anyhow::anyhow!("Invalid attribute index: {}", e))
+            })
+            .collect()
+    }
+
+    input
+        .split(';')
+        .map(|dep| {
+            let parts: Vec<&str> = dep.trim().split('>').collect();
+            if parts.len() != 2 {
+                bail!(
+                    "Invalid dependency format: expected 'lhs>rhs', got '{}'",
+                    dep.trim()
+                );
+            }
+            let lhs = parse_dependency_side(parts[0])?;
+            let rhs = parse_dependency_side(parts[1])?;
+            Ok((lhs, rhs))
+        })
+        .collect()
+}
+
 fn validate_comparative_containment_sets(
     family_name: &str,
     flag: &str,
@@ -3240,6 +3307,8 @@ mod tests {
             deadline: None,
             num_processors: None,
             alphabet_size: None,
+            dependencies: None,
+            num_attributes: None,
             source_string: None,
             target_string: None,
             schedules: None,

problemreductions-cli/tests/cli_tests.rs

@@ -1361,6 +1361,73 @@ fn test_create_set_basis_rejects_out_of_range_elements() {
     assert!(!stderr.contains("panicked at"), "stderr: {stderr}");
 }
 
+#[test]
+fn test_create_minimum_cardinality_key_problem_help_uses_supported_flags() {
+    let output = pred()
+        .args(["create", "MinimumCardinalityKey"])
+        .output()
+        .unwrap();
+    assert!(!output.status.success());
+    let stderr = String::from_utf8_lossy(&output.stderr);
+    assert!(stderr.contains("--num-attributes"), "stderr: {stderr}");
+    assert!(stderr.contains("--dependencies"), "stderr: {stderr}");
+    assert!(stderr.contains("--k"), "stderr: {stderr}");
+    assert!(
+        stderr.contains("semicolon-separated dependencies"),
+        "stderr: {stderr}"
+    );
+    assert!(!stderr.contains("--bound-k"), "stderr: {stderr}");
+}
+
+#[test]
+fn test_create_minimum_cardinality_key_allows_empty_lhs_dependency() {
+    let output = pred()
+        .args([
+            "create",
+            "MinimumCardinalityKey",
+            "--num-attributes",
+            "1",
+            "--dependencies",
+            ">0",
+            "--k",
+            "1",
+        ])
+        .output()
+        .unwrap();
+    assert!(
+        output.status.success(),
+        "stderr: {}",
+        String::from_utf8_lossy(&output.stderr)
+    );
+
+    let stdout = String::from_utf8(output.stdout).unwrap();
+    let json: serde_json::Value = serde_json::from_str(&stdout).unwrap();
+    assert_eq!(json["type"], "MinimumCardinalityKey");
+    assert_eq!(json["data"]["num_attributes"], 1);
+    assert_eq!(json["data"]["bound_k"], 1);
+    assert_eq!(json["data"]["dependencies"][0][0], serde_json::json!([]));
+    assert_eq!(json["data"]["dependencies"][0][1], serde_json::json!([0]));
+}
+
+#[test]
+fn test_create_minimum_cardinality_key_missing_num_attributes_message() {
+    let output = pred()
+        .args([
+            "create",
+            "MinimumCardinalityKey",
+            "--dependencies",
+            "0>0",
+            "--k",
+            "1",
+        ])
+        .output()
+        .unwrap();
+    assert!(!output.status.success());
+    let stderr = String::from_utf8_lossy(&output.stderr);
+    assert!(stderr.contains("MinimumCardinalityKey requires --num-attributes"));
+    assert!(!stderr.contains("--num-vertices"), "stderr: {stderr}");
+}
+
 #[test]
 fn test_create_then_evaluate() {
     // Create a problem

src/example_db/fixtures/examples.json

@@ -24,6 +24,7 @@
     {"problem":"MaximumIndependentSet","variant":{"graph":"SimpleGraph","weight":"i32"},"instance":{"graph":{"inner":{"edge_property":"undirected","edges":[[0,1,null],[1,2,null],[2,3,null],[3,4,null],[4,0,null],[5,7,null],[7,9,null],[9,6,null],[6,8,null],[8,5,null],[0,5,null],[1,6,null],[2,7,null],[3,8,null],[4,9,null]],"node_holes":[],"nodes":[null,null,null,null,null,null,null,null,null,null]}},"weights":[5,1,1,1,1,3,1,1,1,3]},"samples":[{"config":[1,0,1,0,0,0,0,0,1,1],"metric":{"Valid":10}}],"optimal":[{"config":[1,0,1,0,0,0,0,0,1,1],"metric":{"Valid":10}}]},
     {"problem":"MaximumMatching","variant":{"graph":"SimpleGraph","weight":"i32"},"instance":{"edge_weights":[1,1,1,1,1,1],"graph":{"inner":{"edge_property":"undirected","edges":[[0,1,null],[0,2,null],[1,3,null],[2,3,null],[2,4,null],[3,4,null]],"node_holes":[],"nodes":[null,null,null,null,null]}}},"samples":[{"config":[1,0,0,0,1,0],"metric":{"Valid":2}}],"optimal":[{"config":[0,0,1,0,1,0],"metric":{"Valid":2}},{"config":[0,1,0,0,0,1],"metric":{"Valid":2}},{"config":[0,1,1,0,0,0],"metric":{"Valid":2}},{"config":[1,0,0,0,0,1],"metric":{"Valid":2}},{"config":[1,0,0,0,1,0],"metric":{"Valid":2}},{"config":[1,0,0,1,0,0],"metric":{"Valid":2}}]},
     {"problem":"MaximumSetPacking","variant":{"weight":"i32"},"instance":{"sets":[[0,1],[1,2],[2,3],[3,4]],"weights":[1,1,1,1]},"samples":[{"config":[1,0,1,0],"metric":{"Valid":2}}],"optimal":[{"config":[0,1,0,1],"metric":{"Valid":2}},{"config":[1,0,0,1],"metric":{"Valid":2}},{"config":[1,0,1,0],"metric":{"Valid":2}}]},
+    {"problem":"MinimumCardinalityKey","variant":{},"instance":{"bound_k":2,"dependencies":[[[0,1],[2]],[[0,2],[3]],[[1,3],[4]],[[2,4],[5]]],"num_attributes":6},"samples":[{"config":[1,1,0,0,0,0],"metric":true}],"optimal":[{"config":[1,1,0,0,0,0],"metric":true}]},
     {"problem":"MinimumDominatingSet","variant":{"graph":"SimpleGraph","weight":"i32"},"instance":{"graph":{"inner":{"edge_property":"undirected","edges":[[0,1,null],[0,2,null],[1,3,null],[2,3,null],[2,4,null],[3,4,null]],"node_holes":[],"nodes":[null,null,null,null,null]}},"weights":[1,1,1,1,1]},"samples":[{"config":[0,0,1,1,0],"metric":{"Valid":2}}],"optimal":[{"config":[0,0,1,1,0],"metric":{"Valid":2}},{"config":[0,1,0,0,1],"metric":{"Valid":2}},{"config":[0,1,0,1,0],"metric":{"Valid":2}},{"config":[0,1,1,0,0],"metric":{"Valid":2}},{"config":[1,0,0,0,1],"metric":{"Valid":2}},{"config":[1,0,0,1,0],"metric":{"Valid":2}},{"config":[1,0,1,0,0],"metric":{"Valid":2}}]},
     {"problem":"MinimumFeedbackVertexSet","variant":{"weight":"i32"},"instance":{"graph":{"inner":{"edge_property":"directed","edges":[[0,1,null],[1,2,null],[2,0,null],[0,3,null],[3,4,null],[4,1,null],[4,2,null]],"node_holes":[],"nodes":[null,null,null,null,null]}},"weights":[1,1,1,1,1]},"samples":[{"config":[1,0,0,0,0],"metric":{"Valid":1}}],"optimal":[{"config":[0,0,1,0,0],"metric":{"Valid":1}},{"config":[1,0,0,0,0],"metric":{"Valid":1}}]},
     {"problem":"MinimumMultiwayCut","variant":{"graph":"SimpleGraph","weight":"i32"},"instance":{"edge_weights":[2,3,1,2,4,5],"graph":{"inner":{"edge_property":"undirected","edges":[[0,1,null],[1,2,null],[2,3,null],[3,4,null],[0,4,null],[1,3,null]],"node_holes":[],"nodes":[null,null,null,null,null]}},"terminals":[0,2,4]},"samples":[{"config":[1,0,0,1,1,0],"metric":{"Valid":8}}],"optimal":[{"config":[1,0,0,1,1,0],"metric":{"Valid":8}}]},

src/models/mod.rs

@@ -27,5 +27,6 @@ pub use misc::{
     ShortestCommonSupersequence, StaffScheduling, StringToStringCorrection, SubsetSum,
 };
 pub use set::{
-    ComparativeContainment, ExactCoverBy3Sets, MaximumSetPacking, MinimumSetCovering, SetBasis,
+    ComparativeContainment, ExactCoverBy3Sets, MaximumSetPacking, MinimumCardinalityKey,
+    MinimumSetCovering, SetBasis,
 };