Fix #445: [Model] AdditionalKey

docs/paper/reductions.typ

@@ -62,6 +62,7 @@
 
 // Problem display names for theorem headers
 #let display-name = (
+  "AdditionalKey": [Additional Key],
   "MaximumIndependentSet": [Maximum Independent Set],
   "MinimumVertexCover": [Minimum Vertex Cover],
   "MaxCut": [Max-Cut],
@@ -3638,6 +3639,29 @@ A classical NP-complete problem from Garey and Johnson @garey1979[Ch.~3, p.~76],
   ) <fig:d2cif>
 ]
 
+#problem-def("AdditionalKey")[
+  Given a set $A$ of attribute names, a collection $F$ of functional dependencies on $A$,
+  a subset $R subset.eq A$, and a set $K$ of candidate keys for the relational scheme $chevron.l R, F chevron.r$,
+  determine whether there exists a subset $R' subset.eq R$ such that $R' in.not K$,
+  the closure $R'^+$ under $F$ equals $R$, and no proper subset of $R'$ also has this property.
+][
+  A classical NP-complete problem from relational database theory @beeri1979.
+  Enumerating all candidate keys is necessary to verify Boyce-Codd Normal Form (BCNF),
+  and the NP-completeness of Additional Key implies that BCNF testing is intractable in general.
+  The best known exact algorithm is brute-force enumeration of all $2^(|R|)$ subsets,
+  checking each for the key property via closure computation under Armstrong's axioms.
+  #footnote[No algorithm improving on brute-force is known for the Additional Key problem.]
+
+  *Example.* Consider attribute set $A = {0, 1, 2, 3, 4, 5}$ with functional dependencies
+  $F = {{0,1} -> {2,3}, {2,3} -> {4,5}, {4,5} -> {0,1}, {0,2} -> {3}, {3,5} -> {1}}$,
+  relation $R = A$, and known keys $K = {{0,1}, {2,3}, {4,5}}$.
+  The subset ${0,2}$ is an additional key: starting from ${0,2}$, we apply ${0,2} -> {3}$
+  to get ${0,2,3}$, then ${2,3} -> {4,5}$ to get ${0,2,3,4,5}$, then ${4,5} -> {0,1}$
+  to reach $R^+ = A$. The set ${0,2}$ is minimal (neither ${0}$ nor ${2}$ alone determines $A$)
+  and ${0,2} in.not K$, so the answer is YES.
+]
+
+
 #{
   let x = load-model-example("ConjunctiveBooleanQuery")
   let d = x.instance.domain_size

docs/paper/references.bib

@@ -198,6 +198,17 @@ @article{lucas2014
   year    = {2014}
 }
 
+@article{beeri1979,
+  author  = {Catriel Beeri and Philip A. Bernstein},
+  title   = {Computational Problems Related to the Design of Normal Form Relational Schemas},
+  journal = {ACM Transactions on Database Systems},
+  volume  = {4},
+  number  = {1},
+  pages   = {30--59},
+  year    = {1979},
+  doi     = {10.1145/320064.320066}
+}
+
 @article{barahona1982,
   author  = {Francisco Barahona},
   title   = {On the computational complexity of Ising spin glass models},

problemreductions-cli/src/cli.rs

@@ -257,6 +257,7 @@ Flags by problem type:
   OptimalLinearArrangement        --graph, --bound
   RuralPostman (RPP)              --graph, --edge-weights, --required-edges, --bound
   MultipleChoiceBranching         --arcs [--weights] --partition --bound [--num-vertices]
+  AdditionalKey                   --num-attributes, --dependencies, --relation-attrs [--known-keys]
   SubgraphIsomorphism             --graph (host), --pattern (pattern)
   LCS                             --strings, --bound [--alphabet-size]
   FAS                             --arcs [--weights] [--num-vertices]
@@ -540,6 +541,18 @@ pub struct CreateArgs {
     /// Alphabet size for LCS, SCS, or StringToStringCorrection (optional; inferred from the input strings if omitted)
     #[arg(long)]
     pub alphabet_size: Option<usize>,
+    /// Number of attributes for AdditionalKey or MinimumCardinalityKey
+    #[arg(long)]
+    pub num_attributes: Option<usize>,
+    /// Functional dependencies for AdditionalKey (e.g., "0,1:2,3;2,3:4,5") or MinimumCardinalityKey (semicolon-separated "lhs>rhs" pairs, e.g., "0,1>2;0,2>3")
+    #[arg(long)]
+    pub dependencies: Option<String>,
+    /// Relation scheme attributes for AdditionalKey (comma-separated, e.g., "0,1,2,3,4,5")
+    #[arg(long)]
+    pub relation_attrs: Option<String>,
+    /// Known candidate keys for AdditionalKey (e.g., "0,1;2,3")
+    #[arg(long)]
+    pub known_keys: Option<String>,
     /// Domain size for ConjunctiveBooleanQuery
     #[arg(long)]
     pub domain_size: Option<usize>,
@@ -558,12 +571,6 @@ pub struct CreateArgs {
     /// Number of groups for SumOfSquaresPartition
     #[arg(long)]
     pub num_groups: 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

@@ -14,7 +14,7 @@ use problemreductions::models::graph::{
     MultipleChoiceBranching, SteinerTree, StrongConnectivityAugmentation,
 };
 use problemreductions::models::misc::{
-    BinPacking, BoyceCoddNormalFormViolation, CbqRelation, ConjunctiveBooleanQuery,
+    AdditionalKey, BinPacking, BoyceCoddNormalFormViolation, CbqRelation, ConjunctiveBooleanQuery,
     FlowShopScheduling, LongestCommonSubsequence, MinimumTardinessSequencing,
     MultiprocessorScheduling, PaintShop, PartiallyOrderedKnapsack, QueryArg,
     RectilinearPictureCompression, ResourceConstrainedScheduling,
@@ -121,6 +121,10 @@ fn all_data_flags_empty(args: &CreateArgs) -> bool {
         && args.sink_2.is_none()
         && args.requirement_1.is_none()
         && args.requirement_2.is_none()
+        && args.num_attributes.is_none()
+        && args.dependencies.is_none()
+        && args.relation_attrs.is_none()
+        && args.known_keys.is_none()
         && args.domain_size.is_none()
         && args.relations.is_none()
         && args.conjuncts_spec.is_none()
@@ -412,6 +416,7 @@ fn example_for(canonical: &str, graph_type: Option<&str>) -> &'static str {
         "MultipleChoiceBranching" => {
             "--arcs \"0>1,0>2,1>3,2>3,1>4,3>5,4>5,2>4\" --weights 3,2,4,1,2,3,1,3 --partition \"0,1;2,3;4,7;5,6\" --bound 10"
         }
+        "AdditionalKey" => "--num-attributes 6 --dependencies \"0,1:2,3;2,3:4,5;4,5:0,1\" --relation-attrs 0,1,2,3,4,5 --known-keys \"0,1;2,3;4,5\"",
         "SubgraphIsomorphism" => "--graph 0-1,1-2,2-0 --pattern 0-1",
         "RectilinearPictureCompression" => {
             "--matrix \"1,1,0,0;1,1,0,0;0,0,1,1;0,0,1,1\" --k 2"
@@ -1402,6 +1407,51 @@ pub fn create(args: &CreateArgs, out: &OutputConfig) -> Result<()> {
             }
         }
 
+        // AdditionalKey
+        "AdditionalKey" => {
+            let usage = "Usage: pred create AdditionalKey --num-attributes 6 --dependencies \"0,1:2,3;2,3:4,5\" --relation-attrs \"0,1,2,3,4,5\" --known-keys \"0,1;2,3\"";
+            let num_attributes = args.num_attributes.ok_or_else(|| {
+                anyhow::anyhow!("AdditionalKey requires --num-attributes\n\n{usage}")
+            })?;
+            let deps_str = args.dependencies.as_deref().ok_or_else(|| {
+                anyhow::anyhow!("AdditionalKey requires --dependencies\n\n{usage}")
+            })?;
+            let ra_str = args.relation_attrs.as_deref().ok_or_else(|| {
+                anyhow::anyhow!("AdditionalKey requires --relation-attrs\n\n{usage}")
+            })?;
+            let dependencies: Vec<(Vec<usize>, Vec<usize>)> = deps_str
+                .split(';')
+                .map(|dep| {
+                    let parts: Vec<&str> = dep.trim().split(':').collect();
+                    anyhow::ensure!(
+                        parts.len() == 2,
+                        "Invalid dependency format '{}', expected 'lhs:rhs' (e.g., '0,1:2,3')",
+                        dep.trim()
+                    );
+                    let lhs: Vec<usize> = util::parse_comma_list(parts[0].trim())?;
+                    let rhs: Vec<usize> = util::parse_comma_list(parts[1].trim())?;
+                    Ok((lhs, rhs))
+                })
+                .collect::<Result<Vec<_>>>()?;
+            let relation_attrs: Vec<usize> = util::parse_comma_list(ra_str)?;
+            let known_keys: Vec<Vec<usize>> = match args.known_keys.as_deref() {
+                Some(s) if !s.is_empty() => s
+                    .split(';')
+                    .map(|k| util::parse_comma_list(k.trim()))
+                    .collect::<Result<Vec<_>>>()?,
+                _ => vec![],
+            };
+            (
+                ser(AdditionalKey::new(
+                    num_attributes,
+                    dependencies,
+                    relation_attrs,
+                    known_keys,
+                ))?,
+                resolved_variant.clone(),
+            )
+        }
+
         // SubsetSum
         "SubsetSum" => {
             let sizes_str = args.sizes.as_deref().ok_or_else(|| {
@@ -4439,6 +4489,8 @@ mod tests {
             domain_size: None,
             relations: None,
             conjuncts_spec: None,
+            relation_attrs: None,
+            known_keys: None,
             costs: None,
             cut_bound: None,
             size_bound: None,

src/lib.rs

@@ -60,13 +60,14 @@ pub mod prelude {
         UndirectedTwoCommodityIntegralFlow,
     };
     pub use crate::models::misc::{
-        BinPacking, BoyceCoddNormalFormViolation, CbqRelation, ConjunctiveBooleanQuery,
-        ConjunctiveQueryFoldability, Factoring, FlowShopScheduling, Knapsack,
-        LongestCommonSubsequence, MinimumTardinessSequencing, MultiprocessorScheduling, PaintShop,
-        Partition, QueryArg, RectilinearPictureCompression, ResourceConstrainedScheduling,
-        SequencingToMinimizeMaximumCumulativeCost, SequencingWithReleaseTimesAndDeadlines,
-        SequencingWithinIntervals, ShortestCommonSupersequence, StaffScheduling,
-        StringToStringCorrection, SubsetSum, SumOfSquaresPartition, Term,
+        AdditionalKey, BinPacking, BoyceCoddNormalFormViolation, CbqRelation,
+        ConjunctiveBooleanQuery, ConjunctiveQueryFoldability, Factoring, FlowShopScheduling,
+        Knapsack, LongestCommonSubsequence, MinimumTardinessSequencing, MultiprocessorScheduling,
+        PaintShop, Partition, QueryArg, RectilinearPictureCompression,
+        ResourceConstrainedScheduling, SequencingToMinimizeMaximumCumulativeCost,
+        SequencingWithReleaseTimesAndDeadlines, SequencingWithinIntervals,
+        ShortestCommonSupersequence, StaffScheduling, StringToStringCorrection, SubsetSum,
+        SumOfSquaresPartition, Term,
     };
     pub use crate::models::set::{
         ComparativeContainment, ConsecutiveSets, ExactCoverBy3Sets, MaximumSetPacking,