Fix #409: [Model] RootedTreeStorageAssignment (#749)

* Add plan for #409: [Model] RootedTreeStorageAssignment

* Implement #409: [Model] RootedTreeStorageAssignment

* chore: remove plan file after implementation

* chore: fix formatting after merge with main

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

@@ -158,6 +158,7 @@
   "QuantifiedBooleanFormulas": [Quantified Boolean Formulas (QBF)],
   "RectilinearPictureCompression": [Rectilinear Picture Compression],
   "ResourceConstrainedScheduling": [Resource Constrained Scheduling],
+  "RootedTreeStorageAssignment": [Rooted Tree Storage Assignment],
   "SchedulingWithIndividualDeadlines": [Scheduling With Individual Deadlines],
   "SequencingToMinimizeMaximumCumulativeCost": [Sequencing to Minimize Maximum Cumulative Cost],
   "SequencingToMinimizeWeightedCompletionTime": [Sequencing to Minimize Weighted Completion Time],
@@ -2912,6 +2913,70 @@ A classical NP-complete problem from Garey and Johnson @garey1979[Ch.~3, p.~76],
   ]
 }
 
+#{
+  let x = load-model-example("RootedTreeStorageAssignment")
+  let n = x.instance.universe_size
+  let subsets = x.instance.subsets
+  let m = subsets.len()
+  let K = x.instance.bound
+  let config = x.optimal_config
+  let edges = config.enumerate().filter(((v, p)) => v != p).map(((v, p)) => (p, v))
+  let fmt-set(s) = "${" + s.map(e => str(e)).join(", ") + "}$"
+  let highlight-nodes = (0, 2, 4)
+  let highlight-edges = ((0, 2), (2, 4))
+  [
+    #problem-def("RootedTreeStorageAssignment")[
+      Given a finite set $X = {0, 1, dots, #(n - 1)}$, a collection $cal(C) = {X_1, dots, X_m}$ of subsets of $X$, and a nonnegative integer $K$, find a directed rooted tree $T = (X, A)$ and supersets $X_i' supset.eq X_i$ such that every $X_i'$ forms a directed path in $T$ and $sum_(i = 1)^m |X_i' backslash X_i| <= K$.
+    ][
+    Rooted Tree Storage Assignment is the storage-and-retrieval problem SR5 in Garey and Johnson @garey1979. Their catalog credits a reduction from Rooted Tree Arrangement, framing the problem as hierarchical file organization: pick a rooted tree on the records so every request set can be completed to a single root-to-leaf path using only a limited number of extra records. The implementation here uses one parent variable per element of $X$, so the direct exhaustive bound is $|X|^(|X|)$ candidate parent arrays, filtered down to valid rooted trees#footnote[No exact algorithm improving on the direct parent-array search bound is claimed here for the general formulation.].
+
+    *Example.* Let $X = {0, 1, dots, #(n - 1)}$, $K = #K$, and $cal(C) = {#range(m).map(i => $X_#(i + 1)$).join(", ")}$ with #subsets.enumerate().map(((i, s)) => $X_#(i + 1) = #fmt-set(s)$).join(", "). The satisfying parent array $p = (#config.map(str).join(", "))$ encodes the rooted tree with arcs #edges.map(((u, v)) => $(#u, #v)$).join(", "). In this tree, $X_1 = {0, 2}$, $X_2 = {1, 3}$, and $X_4 = {2, 4}$ are already directed paths. The only extension is $X_3 = {0, 4}$, which becomes $X_3' = {0, 2, 4}$ along the path $0 -> 2 -> 4$, so the total extension cost is exactly $1 = K$.
+
+    #pred-commands(
+      "pred create --example " + problem-spec(x) + " -o rooted-tree-storage-assignment.json",
+      "pred solve rooted-tree-storage-assignment.json --solver brute-force",
+      "pred evaluate rooted-tree-storage-assignment.json --config " + x.optimal_config.map(str).join(","),
+    )
+
+    #figure(
+      canvas(length: 1cm, {
+        import draw: *
+
+        let positions = (
+          (1.5, 1.8),
+          (0.6, 0.9),
+          (2.4, 0.9),
+          (0.6, 0.0),
+          (2.4, 0.0),
+        )
+
+        for (u, v) in edges {
+          let highlighted = highlight-edges.contains((u, v))
+          line(
+            positions.at(u),
+            positions.at(v),
+            stroke: if highlighted { 1.2pt + graph-colors.at(0) } else { 0.8pt + luma(140) },
+            mark: (end: "straight", scale: 0.45),
+          )
+        }
+
+        for (vertex, pos) in positions.enumerate() {
+          let highlighted = highlight-nodes.contains(vertex)
+          circle(
+            pos,
+            radius: 0.2,
+            fill: if highlighted { graph-colors.at(0) } else { white },
+            stroke: 0.6pt + black,
+          )
+          content(pos, if highlighted { text(fill: white)[$#vertex$] } else { [$#vertex$] })
+        }
+      }),
+      caption: [Rooted Tree Storage Assignment example. The rooted tree encoded by $p = (#config.map(str).join(", "))$ is shown; the blue path $0 -> 2 -> 4$ is the unique extension needed to realize $X_3 = {0, 4}$ within total cost $K = #K$.],
+    ) <fig:rooted-tree-storage-assignment>
+    ]
+  ]
+}
+
 #{
   let x = load-model-example("TwoDimensionalConsecutiveSets")
   let n = x.instance.alphabet_size

problemreductions-cli/src/cli.rs

@@ -260,6 +260,7 @@ Flags by problem type:
   SetBasis                        --universe, --sets, --k
   MinimumCardinalityKey           --num-attributes, --dependencies, --k
   PrimeAttributeName              --universe, --deps, --query
+  RootedTreeStorageAssignment     --universe, --sets, --bound
   TwoDimensionalConsecutiveSets   --alphabet-size, --sets
   BicliqueCover                   --left, --right, --biedges, --k
   BalancedCompleteBipartiteSubgraph --left, --right, --biedges, --k

problemreductions-cli/src/commands/create.rs

@@ -598,6 +598,7 @@ fn example_for(canonical: &str, graph_type: Option<&str>) -> &'static str {
         "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",
+        "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 --bound 2"
         }
@@ -2668,6 +2669,31 @@ pub fn create(args: &CreateArgs, out: &OutputConfig) -> Result<()> {
             )
         }
 
+        // RootedTreeStorageAssignment
+        "RootedTreeStorageAssignment" => {
+            let usage =
+                "Usage: pred create RootedTreeStorageAssignment --universe 5 --sets \"0,2;1,3;0,4;2,4\" --bound 1";
+            let universe_size = args.universe.ok_or_else(|| {
+                anyhow::anyhow!("RootedTreeStorageAssignment requires --universe\n\n{usage}")
+            })?;
+            let subsets = parse_sets(args)?;
+            let bound = args.bound.ok_or_else(|| {
+                anyhow::anyhow!("RootedTreeStorageAssignment requires --bound\n\n{usage}")
+            })?;
+            let bound = parse_nonnegative_usize_bound(bound, "RootedTreeStorageAssignment", usage)?;
+            (
+                ser(
+                    problemreductions::models::set::RootedTreeStorageAssignment::try_new(
+                        universe_size,
+                        subsets,
+                        bound,
+                    )
+                    .map_err(anyhow::Error::msg)?,
+                )?,
+                resolved_variant.clone(),
+            )
+        }
+
         // BicliqueCover
         "BicliqueCover" => {
             let usage = "pred create BicliqueCover --left 2 --right 2 --biedges 0-0,0-1,1-1 --k 2";
@@ -7495,6 +7521,38 @@ mod tests {
         assert!(err.contains("ExpectedRetrievalCost requires --latency-bound"));
     }
 
+    #[test]
+    fn test_create_rooted_tree_storage_assignment_json() {
+        let mut args = empty_args();
+        args.problem = Some("RootedTreeStorageAssignment".to_string());
+        args.universe = Some(5);
+        args.sets = Some("0,2;1,3;0,4;2,4".to_string());
+        args.bound = Some(1);
+
+        let output_path =
+            std::env::temp_dir().join("pred_test_create_rooted_tree_storage_assignment.json");
+        let out = OutputConfig {
+            output: Some(output_path.clone()),
+            quiet: true,
+            json: false,
+            auto_json: false,
+        };
+
+        create(&args, &out).unwrap();
+
+        let content = std::fs::read_to_string(&output_path).unwrap();
+        let json: serde_json::Value = serde_json::from_str(&content).unwrap();
+        assert_eq!(json["type"], "RootedTreeStorageAssignment");
+        assert_eq!(json["data"]["universe_size"], 5);
+        assert_eq!(
+            json["data"]["subsets"],
+            serde_json::json!([[0, 2], [1, 3], [0, 4], [2, 4]])
+        );
+        assert_eq!(json["data"]["bound"], 1);
+
+        std::fs::remove_file(output_path).ok();
+    }
+
     #[test]
     fn test_create_stacker_crane_json() {
         let mut args = empty_args();

src/lib.rs

@@ -80,7 +80,8 @@ pub mod prelude {
     };
     pub use crate::models::set::{
         ComparativeContainment, ConsecutiveSets, ExactCoverBy3Sets, MaximumSetPacking,
-        MinimumCardinalityKey, MinimumHittingSet, MinimumSetCovering, PrimeAttributeName, SetBasis,
+        MinimumCardinalityKey, MinimumHittingSet, MinimumSetCovering, PrimeAttributeName,
+        RootedTreeStorageAssignment, SetBasis,
     };
 
     // Core traits

src/models/mod.rs

@@ -48,6 +48,6 @@ pub use misc::{
 };
 pub use set::{
     ComparativeContainment, ConsecutiveSets, ExactCoverBy3Sets, MaximumSetPacking,
-    MinimumCardinalityKey, MinimumHittingSet, MinimumSetCovering, PrimeAttributeName, SetBasis,
-    TwoDimensionalConsecutiveSets,
+    MinimumCardinalityKey, MinimumHittingSet, MinimumSetCovering, PrimeAttributeName,
+    RootedTreeStorageAssignment, SetBasis, TwoDimensionalConsecutiveSets,
 };

src/models/set/mod.rs

@@ -8,6 +8,7 @@
 //! - [`MinimumHittingSet`]: Minimum-size universe subset hitting every set
 //! - [`MinimumSetCovering`]: Minimum weight set cover
 //! - [`PrimeAttributeName`]: Determine if an attribute belongs to any candidate key
+//! - [`RootedTreeStorageAssignment`]: Extend subsets to directed tree paths within a total-cost bound
 
 pub(crate) mod comparative_containment;
 pub(crate) mod consecutive_sets;
@@ -17,6 +18,7 @@ pub(crate) mod minimum_cardinality_key;
 pub(crate) mod minimum_hitting_set;
 pub(crate) mod minimum_set_covering;
 pub(crate) mod prime_attribute_name;
+pub(crate) mod rooted_tree_storage_assignment;
 pub(crate) mod set_basis;
 pub(crate) mod two_dimensional_consecutive_sets;
 
@@ -28,6 +30,7 @@ pub use minimum_cardinality_key::MinimumCardinalityKey;
 pub use minimum_hitting_set::MinimumHittingSet;
 pub use minimum_set_covering::MinimumSetCovering;
 pub use prime_attribute_name::PrimeAttributeName;
+pub use rooted_tree_storage_assignment::RootedTreeStorageAssignment;
 pub use set_basis::SetBasis;
 pub use two_dimensional_consecutive_sets::TwoDimensionalConsecutiveSets;
 
@@ -42,6 +45,7 @@ pub(crate) fn canonical_model_example_specs() -> Vec<crate::example_db::specs::M
     specs.extend(minimum_hitting_set::canonical_model_example_specs());
     specs.extend(minimum_set_covering::canonical_model_example_specs());
     specs.extend(prime_attribute_name::canonical_model_example_specs());
+    specs.extend(rooted_tree_storage_assignment::canonical_model_example_specs());
     specs.extend(set_basis::canonical_model_example_specs());
     specs.extend(two_dimensional_consecutive_sets::canonical_model_example_specs());
     specs