Fix #400: [Model] SetBasis

docs/paper/reductions.typ

@@ -75,6 +75,7 @@
   "MaximumClique": [Maximum Clique],
   "MaximumSetPacking": [Maximum Set Packing],
   "MinimumSetCovering": [Minimum Set Covering],
+  "SetBasis": [Set Basis],
   "SpinGlass": [Spin Glass],
   "QUBO": [QUBO],
   "ILP": [Integer Linear Programming],
@@ -1107,6 +1108,43 @@ NP-completeness was established by Garey, Johnson, and Stockmeyer @gareyJohnsonS
   ]
 }
 
+#{
+  let x = load-model-example("SetBasis")
+  let coll = x.instance.collection
+  let m = coll.len()
+  let U-size = x.instance.universe_size
+  let k = x.instance.k
+  let sample = x.samples.at(0)
+  let sat-count = x.optimal.len()
+  let basis = range(k).map(i =>
+    range(U-size).filter(j => sample.config.at(i * U-size + j) == 1)
+  )
+  let fmt-set(s) = "${" + s.map(e => str(e + 1)).join(", ") + "}$"
+  [
+    #problem-def("SetBasis")[
+      Given finite set $S$, collection $cal(C)$ of subsets of $S$, and integer $k$, does there exist a family $cal(B) = {B_1, ..., B_k}$ with each $B_i subset.eq S$ such that for every $C in cal(C)$ there exists $cal(B)_C subset.eq cal(B)$ with $union.big_(B in cal(B)_C) B = C$?
+    ][
+    The Set Basis problem was shown NP-complete by Stockmeyer @stockmeyer1975setbasis and appears as SP7 in Garey & Johnson @garey1979. It asks for an exact union-based description of a family of sets, unlike Set Cover which only requires covering the underlying universe. Applications include data compression, database schema design, and Boolean function minimization. The library's decision encoding uses $k |S|$ membership bits, so brute-force over those bits gives an $O^*(2^(k |S|))$ exact algorithm#footnote[This is the direct search bound induced by the encoding implemented here; we are not aware of a faster general exact worst-case algorithm for this representation.].
+
+    *Example.* Let $S = {1, 2, 3, 4}$, $k = #k$, and $cal(C) = {#range(m).map(i => $C_#(i + 1)$).join(", ")}$ with #coll.enumerate().map(((i, s)) => $C_#(i + 1) = #fmt-set(s)$).join(", "). The sample basis from the issue is $cal(B) = {#range(k).map(i => $B_#(i + 1)$).join(", ")}$ with #basis.enumerate().map(((i, s)) => $B_#(i + 1) = #fmt-set(s)$).join(", "). Then $C_1 = B_1 union B_2$, $C_2 = B_2 union B_3$, $C_3 = B_1 union B_3$, and $C_4 = B_1 union B_2 union B_3$. There are #sat-count satisfying encodings in total: the singleton basis can be permuted in $3! = 6$ ways, and the three pair sets $C_1, C_2, C_3$ also form a basis with another six row permutations.
+
+    #figure(
+      canvas(length: 1cm, {
+        let elems = ((-0.9, 0.2), (0.0, -0.5), (0.9, 0.2), (1.8, -0.5))
+        for i in range(k) {
+          let positions = basis.at(i).map(e => elems.at(e))
+          sregion(positions, pad: 0.28, label: [$B_#(i + 1)$], ..sregion-selected)
+        }
+        for (idx, pos) in elems.enumerate() {
+          selem(pos, label: [#(idx + 1)], fill: if idx < 3 { black } else { luma(160) })
+        }
+      }),
+      caption: [Set Basis example: the singleton basis $cal(B) = {#range(k).map(i => $B_#(i + 1)$).join(", ")}$ reconstructs every target set in $cal(C)$; element $4$ is unused by the target family.],
+    ) <fig:set-basis>
+    ]
+  ]
+}
+
 == Optimization Problems
 
 #{

docs/paper/references.bib

@@ -562,6 +562,15 @@ @article{arora2009
   doi     = {10.1145/1502793.1502794}
 }
 
+@techreport{stockmeyer1975setbasis,
+  author      = {Larry J. Stockmeyer},
+  title       = {The Set Basis Problem Is NP-Complete},
+  institution = {IBM Thomas J. Watson Research Center},
+  number      = {RC 5431},
+  address     = {Yorktown Heights, New York},
+  year        = {1975}
+}
+
 @article{cygan2014,
   author  = {Marek Cygan and Daniel Lokshtanov and Marcin Pilipczuk and Micha{\l} Pilipczuk and Saket Saurabh},
   title   = {Minimum Bisection Is Fixed Parameter Tractable},

docs/src/reductions/problem_schemas.json

@@ -594,6 +594,27 @@
       }
     ]
   },
+  {
+    "name": "SetBasis",
+    "description": "Determine whether a collection of sets admits a basis of size k under union",
+    "fields": [
+      {
+        "name": "universe_size",
+        "type_name": "usize",
+        "description": "Size of the ground set S"
+      },
+      {
+        "name": "collection",
+        "type_name": "Vec<Vec<usize>>",
+        "description": "Collection C of target subsets of S"
+      },
+      {
+        "name": "k",
+        "type_name": "usize",
+        "description": "Required number of basis sets"
+      }
+    ]
+  },
   {
     "name": "ShortestCommonSupersequence",
     "description": "Find a common supersequence of bounded length for a set of strings",

docs/src/reductions/reduction_graph.json

@@ -464,6 +464,13 @@
       "doc_path": "models/formula/struct.Satisfiability.html",
       "complexity": "2^num_variables"
     },
+    {
+      "name": "SetBasis",
+      "variant": {},
+      "category": "set",
+      "doc_path": "models/set/struct.SetBasis.html",
+      "complexity": "2^(basis_size * universe_size)"
+    },
     {
       "name": "ShortestCommonSupersequence",
       "variant": {},
@@ -569,7 +576,7 @@
     },
     {
       "source": 4,
-      "target": 54,
+      "target": 55,
       "overhead": [
         {
           "field": "num_spins",
@@ -733,7 +740,7 @@
     },
     {
       "source": 21,
-      "target": 58,
+      "target": 59,
       "overhead": [
         {
           "field": "num_elements",
@@ -789,7 +796,7 @@
     },
     {
       "source": 25,
-      "target": 54,
+      "target": 55,
       "overhead": [
         {
           "field": "num_spins",
@@ -1235,7 +1242,7 @@
     },
     {
       "source": 49,
-      "target": 53,
+      "target": 54,
       "overhead": [
         {
           "field": "num_spins",
@@ -1320,7 +1327,7 @@
       "doc_path": "rules/sat_minimumdominatingset/index.html"
     },
     {
-      "source": 53,
+      "source": 54,
       "target": 49,
       "overhead": [
         {
@@ -1331,7 +1338,7 @@
       "doc_path": "rules/spinglass_qubo/index.html"
     },
     {
-      "source": 54,
+      "source": 55,
       "target": 25,
       "overhead": [
         {
@@ -1346,8 +1353,8 @@
       "doc_path": "rules/spinglass_maxcut/index.html"
     },
     {
-      "source": 54,
-      "target": 53,
+      "source": 55,
+      "target": 54,
       "overhead": [
         {
           "field": "num_spins",
@@ -1361,7 +1368,7 @@
       "doc_path": "rules/spinglass_casts/index.html"
     },
     {
-      "source": 59,
+      "source": 60,
       "target": 12,
       "overhead": [
         {
@@ -1376,7 +1383,7 @@
       "doc_path": "rules/travelingsalesman_ilp/index.html"
     },
     {
-      "source": 59,
+      "source": 60,
       "target": 49,
       "overhead": [
         {

problemreductions-cli/src/cli.rs

@@ -231,6 +231,7 @@ Flags by problem type:
   MaximumSetPacking               --sets [--weights]
   MinimumSetCovering              --universe, --sets [--weights]
   X3C (ExactCoverBy3Sets)         --universe, --sets (3 elements each)
+  SetBasis                        --universe, --sets, --k
   BicliqueCover                   --left, --right, --biedges, --k
   BMF                             --matrix (0/1), --rank
   SteinerTree                     --graph, --edge-weights, --terminals
@@ -264,7 +265,8 @@ Examples:
   pred create MIS/UnitDiskGraph --positions \"0,0;1,0;0.5,0.8\" --radius 1.5
   pred create MIS --random --num-vertices 10 --edge-prob 0.3
   pred create FVS --arcs \"0>1,1>2,2>0\" --weights 1,1,1
-  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 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")]
 pub struct CreateArgs {
     /// Problem type (e.g., MIS, QUBO, SAT). Omit when using --example.
     #[arg(value_parser = crate::problem_name::ProblemNameParser)]

problemreductions-cli/src/commands/create.rs

@@ -213,6 +213,21 @@ fn type_format_hint(type_name: &str, graph_type: Option<&str>) -> &'static str {
     }
 }
 
+fn cli_flag_name(field_name: &str) -> String {
+    match field_name {
+        "universe_size" => "universe".to_string(),
+        "collection" | "subsets" => "sets".to_string(),
+        "left_size" => "left".to_string(),
+        "right_size" => "right".to_string(),
+        "edges" => "biedges".to_string(),
+        "vertex_weights" => "weights".to_string(),
+        "edge_lengths" => "edge-weights".to_string(),
+        "num_tasks" => "n".to_string(),
+        "precedences" => "precedence-pairs".to_string(),
+        _ => field_name.replace('_', "-"),
+    }
+}
+
 fn example_for(canonical: &str, graph_type: Option<&str>) -> &'static str {
     match canonical {
         "MaximumIndependentSet"
@@ -248,6 +263,7 @@ fn example_for(canonical: &str, graph_type: Option<&str>) -> &'static str {
         }
         "SubgraphIsomorphism" => "--graph 0-1,1-2,2-0 --pattern 0-1",
         "SubsetSum" => "--sizes 3,7,1,8,2,4 --target 11",
+        "SetBasis" => "--universe 4 --sets \"0,1;1,2;0,2;0,1,2\" --k 3",
         "ShortestCommonSupersequence" => "--strings \"0,1,2;1,2,0\" --bound 4",
         _ => "",
     }
@@ -280,7 +296,7 @@ fn print_problem_help(canonical: &str, graph_type: Option<&str>) -> Result<()> {
                 let hint = type_format_hint(&field.type_name, graph_type);
                 eprintln!(
                     "  --{:<16} {} ({})",
-                    field.name.replace('_', "-"),
+                    cli_flag_name(&field.name),
                     field.description,
                     hint
                 );
@@ -718,6 +734,41 @@ pub fn create(args: &CreateArgs, out: &OutputConfig) -> Result<()> {
             )
         }
 
+        // SetBasis
+        "SetBasis" => {
+            let universe = args.universe.ok_or_else(|| {
+                anyhow::anyhow!(
+                    "SetBasis requires --universe, --sets, and --k\n\n\
+                     Usage: pred create SetBasis --universe 4 --sets \"0,1;1,2;0,2;0,1,2\" --k 3"
+                )
+            })?;
+            let k = args.k.ok_or_else(|| {
+                anyhow::anyhow!(
+                    "SetBasis requires --k\n\n\
+                     Usage: pred create SetBasis --universe 4 --sets \"0,1;1,2;0,2;0,1,2\" --k 3"
+                )
+            })?;
+            let sets = parse_sets(args)?;
+            for (i, set) in sets.iter().enumerate() {
+                for &element in set {
+                    if element >= universe {
+                        bail!(
+                            "Set {} contains element {} which is outside universe of size {}",
+                            i,
+                            element,
+                            universe
+                        );
+                    }
+                }
+            }
+            (
+                ser(problemreductions::models::set::SetBasis::new(
+                    universe, sets, k,
+                ))?,
+                resolved_variant.clone(),
+            )
+        }
+
         // BicliqueCover
         "BicliqueCover" => {
             let left = args.left.ok_or_else(|| {