Fix #210: [Model] Partition

docs/paper/reductions.typ

@@ -96,6 +96,7 @@
   "LongestCommonSubsequence": [Longest Common Subsequence],
   "ExactCoverBy3Sets": [Exact Cover by 3-Sets],
   "SubsetSum": [Subset Sum],
+  "Partition": [Partition],
   "MinimumFeedbackArcSet": [Minimum Feedback Arc Set],
   "MinimumFeedbackVertexSet": [Minimum Feedback Vertex Set],
   "ShortestCommonSupersequence": [Shortest Common Supersequence],
@@ -1705,6 +1706,14 @@ NP-completeness was established by Garey, Johnson, and Stockmeyer @gareyJohnsonS
   *Example.* Let $A = {3, 7, 1, 8, 2, 4}$ ($n = 6$) and target $B = 11$. Selecting $A' = {3, 8}$ gives sum $3 + 8 = 11 = B$. Another solution: $A' = {7, 4}$ with sum $7 + 4 = 11 = B$.
 ]
 
+#problem-def("Partition")[
+  Given a finite set $A = {a_0, dots, a_(n-1)}$ with sizes $s(a_i) in ZZ^+$, determine whether there exists a subset $A' subset.eq A$ such that $sum_(a in A') s(a) = sum_(a in A without A') s(a)$.
+][
+  One of Karp's 21 NP-complete problems @karp1972, listed as SP12 in Garey & Johnson @garey1979. Partition is the special case of Subset Sum where the target equals half the total sum. Though NP-complete, it is only _weakly_ NP-hard: a dynamic-programming algorithm runs in $O(n dot B_"total")$ pseudo-polynomial time, where $B_"total" = sum_i s(a_i)$. The best known exact algorithm is the $O^*(2^(n slash 2))$ meet-in-the-middle approach of Schroeppel and Shamir (1981).
+
+  *Example.* Let $A = {3, 1, 1, 2, 2, 1}$ ($n = 6$, total sum $= 10$). Setting $A' = {3, 2}$ (indices 0, 3) gives sum $3 + 2 = 5 = 10 slash 2$, and $A without A' = {1, 1, 2, 1}$ also sums to 5. Hence a balanced partition exists.
+]
+
 #{
   let x = load-model-example("ShortestCommonSupersequence")
   let alpha-size = x.instance.alphabet_size

docs/src/reductions/problem_schemas.json

@@ -525,6 +525,17 @@
       }
     ]
   },
+  {
+    "name": "Partition",
+    "description": "Determine whether a multiset of positive integers can be partitioned into two subsets of equal sum",
+    "fields": [
+      {
+        "name": "sizes",
+        "type_name": "Vec<u64>",
+        "description": "Positive integer size for each element"
+      }
+    ]
+  },
   {
     "name": "PartitionIntoTriangles",
     "description": "Partition vertices into triangles (K3 subgraphs)",

docs/src/reductions/reduction_graph.json

@@ -429,6 +429,13 @@
       "doc_path": "models/misc/struct.PaintShop.html",
       "complexity": "2^num_cars"
     },
+    {
+      "name": "Partition",
+      "variant": {},
+      "category": "misc",
+      "doc_path": "models/misc/struct.Partition.html",
+      "complexity": "2^(num_elements / 2)"
+    },
     {
       "name": "PartitionIntoTriangles",
       "variant": {
@@ -569,7 +576,7 @@
     },
     {
       "source": 4,
-      "target": 54,
+      "target": 55,
       "overhead": [
         {
           "field": "num_spins",
@@ -629,7 +636,7 @@
     },
     {
       "source": 12,
-      "target": 49,
+      "target": 50,
       "overhead": [
         {
           "field": "num_vars",
@@ -670,7 +677,7 @@
     },
     {
       "source": 19,
-      "target": 49,
+      "target": 50,
       "overhead": [
         {
           "field": "num_vars",
@@ -696,7 +703,7 @@
     },
     {
       "source": 20,
-      "target": 49,
+      "target": 50,
       "overhead": [
         {
           "field": "num_vars",
@@ -722,7 +729,7 @@
     },
     {
       "source": 21,
-      "target": 49,
+      "target": 50,
       "overhead": [
         {
           "field": "num_vars",
@@ -733,7 +740,7 @@
     },
     {
       "source": 21,
-      "target": 58,
+      "target": 59,
       "overhead": [
         {
           "field": "num_elements",
@@ -744,7 +751,7 @@
     },
     {
       "source": 22,
-      "target": 51,
+      "target": 52,
       "overhead": [
         {
           "field": "num_clauses",
@@ -763,7 +770,7 @@
     },
     {
       "source": 23,
-      "target": 49,
+      "target": 50,
       "overhead": [
         {
           "field": "num_vars",
@@ -789,7 +796,7 @@
     },
     {
       "source": 25,
-      "target": 54,
+      "target": 55,
       "overhead": [
         {
           "field": "num_spins",
@@ -1104,7 +1111,7 @@
     },
     {
       "source": 37,
-      "target": 49,
+      "target": 50,
       "overhead": [
         {
           "field": "num_vars",
@@ -1219,7 +1226,7 @@
       "doc_path": "rules/minimumvertexcover_minimumsetcovering/index.html"
     },
     {
-      "source": 49,
+      "source": 50,
       "target": 12,
       "overhead": [
         {
@@ -1234,8 +1241,8 @@
       "doc_path": "rules/qubo_ilp/index.html"
     },
     {
-      "source": 49,
-      "target": 53,
+      "source": 50,
+      "target": 54,
       "overhead": [
         {
           "field": "num_spins",
@@ -1245,7 +1252,7 @@
       "doc_path": "rules/spinglass_qubo/index.html"
     },
     {
-      "source": 51,
+      "source": 52,
       "target": 4,
       "overhead": [
         {
@@ -1260,7 +1267,7 @@
       "doc_path": "rules/sat_circuitsat/index.html"
     },
     {
-      "source": 51,
+      "source": 52,
       "target": 16,
       "overhead": [
         {
@@ -1275,7 +1282,7 @@
       "doc_path": "rules/sat_coloring/index.html"
     },
     {
-      "source": 51,
+      "source": 52,
       "target": 21,
       "overhead": [
         {
@@ -1290,7 +1297,7 @@
       "doc_path": "rules/sat_ksat/index.html"
     },
     {
-      "source": 51,
+      "source": 52,
       "target": 30,
       "overhead": [
         {
@@ -1305,7 +1312,7 @@
       "doc_path": "rules/sat_maximumindependentset/index.html"
     },
     {
-      "source": 51,
+      "source": 52,
       "target": 39,
       "overhead": [
         {
@@ -1320,8 +1327,8 @@
       "doc_path": "rules/sat_minimumdominatingset/index.html"
     },
     {
-      "source": 53,
-      "target": 49,
+      "source": 54,
+      "target": 50,
       "overhead": [
         {
           "field": "num_vars",
@@ -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": [
         {
@@ -1376,8 +1383,8 @@
       "doc_path": "rules/travelingsalesman_ilp/index.html"
     },
     {
-      "source": 59,
-      "target": 49,
+      "source": 60,
+      "target": 50,
       "overhead": [
         {
           "field": "num_vars",

src/example_db/fixtures/examples.json

@@ -25,6 +25,7 @@
     {"problem":"MinimumTardinessSequencing","variant":{},"instance":{"deadlines":[2,3,1,4],"num_tasks":4,"precedences":[[0,2]]},"samples":[{"config":[0,0,0,0],"metric":{"Valid":1}}],"optimal":[{"config":[0,0,0,0],"metric":{"Valid":1}},{"config":[0,0,1,0],"metric":{"Valid":1}},{"config":[0,1,0,0],"metric":{"Valid":1}},{"config":[0,2,0,0],"metric":{"Valid":1}},{"config":[1,0,0,0],"metric":{"Valid":1}},{"config":[1,0,1,0],"metric":{"Valid":1}},{"config":[3,0,0,0],"metric":{"Valid":1}}]},
     {"problem":"MinimumVertexCover","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":[1,0,0,1,1],"metric":{"Valid":3}}],"optimal":[{"config":[0,1,1,0,1],"metric":{"Valid":3}},{"config":[0,1,1,1,0],"metric":{"Valid":3}},{"config":[1,0,0,1,1],"metric":{"Valid":3}},{"config":[1,0,1,1,0],"metric":{"Valid":3}}]},
     {"problem":"PaintShop","variant":{},"instance":{"car_labels":["A","B","C"],"is_first":[true,true,false,true,false,false],"num_cars":3,"sequence_indices":[0,1,0,2,1,2]},"samples":[{"config":[0,0,1],"metric":{"Valid":2}}],"optimal":[{"config":[0,0,1],"metric":{"Valid":2}},{"config":[0,1,1],"metric":{"Valid":2}},{"config":[1,0,0],"metric":{"Valid":2}},{"config":[1,1,0],"metric":{"Valid":2}}]},
+    {"problem":"Partition","variant":{},"instance":{"sizes":[3,1,1,2,2,1]},"samples":[{"config":[1,0,0,1,0,0],"metric":true},{"config":[0,0,0,0,0,0],"metric":false}],"optimal":[{"config":[0,0,0,1,1,1],"metric":true},{"config":[0,0,1,1,1,0],"metric":true},{"config":[0,1,0,1,1,0],"metric":true},{"config":[0,1,1,0,1,1],"metric":true},{"config":[0,1,1,1,0,1],"metric":true},{"config":[1,0,0,0,1,0],"metric":true},{"config":[1,0,0,1,0,0],"metric":true},{"config":[1,0,1,0,0,1],"metric":true},{"config":[1,1,0,0,0,1],"metric":true},{"config":[1,1,1,0,0,0],"metric":true}]},
     {"problem":"PartitionIntoTriangles","variant":{"graph":"SimpleGraph"},"instance":{"graph":{"inner":{"edge_property":"undirected","edges":[[0,1,null],[0,2,null],[1,2,null],[3,4,null],[3,5,null],[4,5,null],[0,3,null]],"node_holes":[],"nodes":[null,null,null,null,null,null]}}},"samples":[{"config":[0,0,0,1,1,1],"metric":true}],"optimal":[{"config":[0,0,0,1,1,1],"metric":true},{"config":[1,1,1,0,0,0],"metric":true}]},
     {"problem":"QUBO","variant":{"weight":"f64"},"instance":{"matrix":[[-1.0,2.0,0.0],[0.0,-1.0,2.0],[0.0,0.0,-1.0]],"num_vars":3},"samples":[{"config":[1,0,1],"metric":{"Valid":-2.0}}],"optimal":[{"config":[1,0,1],"metric":{"Valid":-2.0}}]},
     {"problem":"Satisfiability","variant":{},"instance":{"clauses":[{"literals":[1,2]},{"literals":[-1,3]},{"literals":[-2,-3]}],"num_vars":3},"samples":[{"config":[1,0,1],"metric":true}],"optimal":[{"config":[0,1,0],"metric":true},{"config":[1,0,1],"metric":true}]},

src/lib.rs

@@ -56,7 +56,7 @@ pub mod prelude {
     };
     pub use crate::models::misc::{
         BinPacking, Factoring, FlowShopScheduling, Knapsack, LongestCommonSubsequence,
-        MinimumTardinessSequencing, PaintShop, ShortestCommonSupersequence, SubsetSum,
+        MinimumTardinessSequencing, PaintShop, Partition, ShortestCommonSupersequence, SubsetSum,
     };
     pub use crate::models::set::{ExactCoverBy3Sets, MaximumSetPacking, MinimumSetCovering};
 

src/models/misc/mod.rs

@@ -8,6 +8,7 @@
 //! - [`LongestCommonSubsequence`]: Longest Common Subsequence
 //! - [`MinimumTardinessSequencing`]: Minimize tardy tasks in single-machine scheduling
 //! - [`PaintShop`]: Minimize color switches in paint shop scheduling
+//! - [`Partition`]: Partition a multiset into two equal-sum subsets
 //! - [`ShortestCommonSupersequence`]: Find a common supersequence of bounded length
 //! - [`SubsetSum`]: Find a subset summing to exactly a target value
 
@@ -18,6 +19,7 @@ mod knapsack;
 mod longest_common_subsequence;
 mod minimum_tardiness_sequencing;
 pub(crate) mod paintshop;
+pub(crate) mod partition;
 pub(crate) mod shortest_common_supersequence;
 mod subset_sum;
 
@@ -28,6 +30,7 @@ pub use knapsack::Knapsack;
 pub use longest_common_subsequence::LongestCommonSubsequence;
 pub use minimum_tardiness_sequencing::MinimumTardinessSequencing;
 pub use paintshop::PaintShop;
+pub use partition::Partition;
 pub use shortest_common_supersequence::ShortestCommonSupersequence;
 pub use subset_sum::SubsetSum;
 
@@ -36,6 +39,7 @@ pub(crate) fn canonical_model_example_specs() -> Vec<crate::example_db::specs::M
     let mut specs = Vec::new();
     specs.extend(factoring::canonical_model_example_specs());
     specs.extend(paintshop::canonical_model_example_specs());
+    specs.extend(partition::canonical_model_example_specs());
     specs.extend(shortest_common_supersequence::canonical_model_example_specs());
     specs.extend(minimum_tardiness_sequencing::canonical_model_example_specs());
     specs