Fix #403: Add SumOfSquaresPartition model

docs/paper/reductions.typ

@@ -120,6 +120,7 @@
   "StaffScheduling": [Staff Scheduling],
   "MultiprocessorScheduling": [Multiprocessor Scheduling],
   "MinimumTardinessSequencing": [Minimum Tardiness Sequencing],
+  "SumOfSquaresPartition": [Sum of Squares Partition],
   "SequencingWithinIntervals": [Sequencing Within Intervals],
   "DirectedTwoCommodityIntegralFlow": [Directed Two-Commodity Integral Flow],
   "StringToStringCorrection": [String-to-String Correction],
@@ -2301,6 +2302,14 @@ 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("SumOfSquaresPartition")[
+  Given a finite set $A = {a_0, dots, a_(n-1)}$ with sizes $s(a_i) in ZZ^+$, a positive integer $K lt.eq |A|$ (number of groups), and a positive integer $J$ (bound), determine whether $A$ can be partitioned into $K$ disjoint sets $A_1, dots, A_K$ such that $sum_(i=1)^K (sum_(a in A_i) s(a))^2 lt.eq J$.
+][
+  Problem SP19 in Garey and Johnson @garey1979. NP-complete in the strong sense, so no pseudo-polynomial time algorithm exists unless $P = NP$. For fixed $K$, a dynamic-programming algorithm runs in $O(n S^(K-1))$ pseudo-polynomial time, where $S = sum s(a)$. The problem remains NP-complete when the exponent 2 is replaced by any fixed rational $alpha > 1$. #footnote[No algorithm improving on brute-force $O(K^n)$ enumeration is known for the general case.] The squared objective penalizes imbalanced partitions, connecting it to variance minimization, load balancing, and $k$-means clustering. Sum of Squares Partition generalizes Partition ($K = 2$, $J = S^2 slash 2$).
+
+  *Example.* Let $A = {5, 3, 8, 2, 7, 1}$ ($n = 6$), $K = 3$ groups, and bound $J = 240$. The partition $A_1 = {8, 1}$, $A_2 = {5, 2}$, $A_3 = {3, 7}$ gives group sums $9, 7, 10$ and sum of squares $81 + 49 + 100 = 230 lt.eq 240 = J$. With a tighter bound $J = 225$, the best achievable partition has group sums ${9, 9, 8}$ yielding $81 + 81 + 64 = 226 > 225$, so the answer is NO.
+]
+
 #{
   let x = load-model-example("SequencingWithReleaseTimesAndDeadlines")
   let n = x.instance.lengths.len()

problemreductions-cli/src/cli.rs

@@ -232,6 +232,7 @@ Flags by problem type:
   Factoring                       --target, --m, --n
   BinPacking                      --sizes, --capacity
   SubsetSum                       --sizes, --target
+  SumOfSquaresPartition           --sizes, --num-groups, --bound
   PaintShop                       --sequence
   MaximumSetPacking               --sets [--weights]
   MinimumSetCovering              --universe, --sets [--weights]
@@ -500,6 +501,9 @@ 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 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>,

problemreductions-cli/src/commands/create.rs

@@ -16,6 +16,7 @@ use problemreductions::models::misc::{
     BinPacking, FlowShopScheduling, LongestCommonSubsequence, MinimumTardinessSequencing,
     MultiprocessorScheduling, PaintShop, SequencingWithReleaseTimesAndDeadlines,
     SequencingWithinIntervals, ShortestCommonSupersequence, StringToStringCorrection, SubsetSum,
+    SumOfSquaresPartition,
 };
 use problemreductions::models::BiconnectivityAugmentation;
 use problemreductions::prelude::*;
@@ -94,6 +95,7 @@ fn all_data_flags_empty(args: &CreateArgs) -> bool {
         && args.requirements.is_none()
         && args.num_workers.is_none()
         && args.alphabet_size.is_none()
+        && args.num_groups.is_none()
         && args.dependencies.is_none()
         && args.num_attributes.is_none()
         && args.source_string.is_none()
@@ -328,6 +330,7 @@ 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",
+        "SumOfSquaresPartition" => "--sizes 5,3,8,2,7,1 --num-groups 3 --bound 240",
         "ComparativeContainment" => {
             "--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"
         }
@@ -1044,6 +1047,34 @@ pub fn create(args: &CreateArgs, out: &OutputConfig) -> Result<()> {
             )
         }
 
+        // SumOfSquaresPartition
+        "SumOfSquaresPartition" => {
+            let sizes_str = args.sizes.as_deref().ok_or_else(|| {
+                anyhow::anyhow!(
+                    "SumOfSquaresPartition requires --sizes, --num-groups, and --bound\n\n\
+                     Usage: pred create SumOfSquaresPartition --sizes 5,3,8,2,7,1 --num-groups 3 --bound 240"
+                )
+            })?;
+            let num_groups = args.num_groups.ok_or_else(|| {
+                anyhow::anyhow!(
+                    "SumOfSquaresPartition requires --num-groups\n\n\
+                     Usage: pred create SumOfSquaresPartition --sizes 5,3,8,2,7,1 --num-groups 3 --bound 240"
+                )
+            })?;
+            let bound = args.bound.ok_or_else(|| {
+                anyhow::anyhow!(
+                    "SumOfSquaresPartition requires --bound\n\n\
+                     Usage: pred create SumOfSquaresPartition --sizes 5,3,8,2,7,1 --num-groups 3 --bound 240"
+                )
+            })?;
+            let sizes: Vec<i64> = util::parse_comma_list(sizes_str)?;
+            (
+                ser(SumOfSquaresPartition::try_new(sizes, num_groups, bound)
+                    .map_err(anyhow::Error::msg)?)?,
+                resolved_variant.clone(),
+            )
+        }
+
         // PaintShop
         "PaintShop" => {
             let seq_str = args.sequence.as_deref().ok_or_else(|| {
@@ -3451,6 +3482,7 @@ mod tests {
             schedules: None,
             requirements: None,
             num_workers: None,
+            num_groups: None,
         }
     }
 

problemreductions-cli/tests/cli_tests.rs

@@ -1361,6 +1361,29 @@ fn test_create_set_basis_rejects_out_of_range_elements() {
     assert!(!stderr.contains("panicked at"), "stderr: {stderr}");
 }
 
+#[test]
+fn test_create_sum_of_squares_partition_rejects_negative_bound_without_panicking() {
+    let output = pred()
+        .args([
+            "create",
+            "SumOfSquaresPartition",
+            "--sizes",
+            "1,2,3",
+            "--num-groups",
+            "2",
+            "--bound=-1",
+        ])
+        .output()
+        .unwrap();
+    assert!(!output.status.success());
+    let stderr = String::from_utf8_lossy(&output.stderr);
+    assert!(
+        stderr.contains("Bound must be nonnegative"),
+        "stderr: {stderr}"
+    );
+    assert!(!stderr.contains("panicked at"), "stderr: {stderr}");
+}
+
 #[test]
 fn test_create_minimum_cardinality_key_problem_help_uses_supported_flags() {
     let output = pred()
@@ -3851,6 +3874,34 @@ fn test_create_pipe_to_solve() {
     );
 }
 
+#[test]
+fn test_solve_ilp_error_suggests_brute_force_fallback() {
+    let problem_json = r#"{
+        "type": "SumOfSquaresPartition",
+        "data": {
+            "sizes": [5, 3, 8, 2, 7, 1],
+            "num_groups": 3,
+            "bound": 240
+        }
+    }"#;
+    let tmp = std::env::temp_dir().join("pred_test_sum_of_squares_partition.json");
+    std::fs::write(&tmp, problem_json).unwrap();
+
+    let output = pred()
+        .args(["solve", tmp.to_str().unwrap()])
+        .output()
+        .unwrap();
+    assert!(!output.status.success());
+
+    let stderr = String::from_utf8_lossy(&output.stderr);
+    assert!(
+        stderr.contains("--solver brute-force"),
+        "stderr should suggest the brute-force fallback, got: {stderr}"
+    );
+
+    std::fs::remove_file(&tmp).ok();
+}
+
 #[test]
 fn test_create_multiple_choice_branching_pipe_to_solve() {
     let create_out = pred()

src/example_db/fixtures/examples.json

@@ -49,6 +49,7 @@
     {"problem":"SteinerTree","variant":{"graph":"SimpleGraph","weight":"i32"},"instance":{"edge_weights":[2,5,2,1,5,6,1],"graph":{"inner":{"edge_property":"undirected","edges":[[0,1,null],[0,3,null],[1,2,null],[1,3,null],[2,3,null],[2,4,null],[3,4,null]],"node_holes":[],"nodes":[null,null,null,null,null]}},"terminals":[0,2,4]},"samples":[{"config":[1,0,1,1,0,0,1],"metric":{"Valid":6}}],"optimal":[{"config":[1,0,1,1,0,0,1],"metric":{"Valid":6}}]},
     {"problem":"StringToStringCorrection","variant":{},"instance":{"alphabet_size":4,"bound":2,"source":[0,1,2,3,1,0],"target":[0,1,3,2,1]},"samples":[{"config":[8,5],"metric":true}],"optimal":[{"config":[5,7],"metric":true},{"config":[8,5],"metric":true}]},
     {"problem":"StrongConnectivityAugmentation","variant":{"weight":"i32"},"instance":{"bound":8,"candidate_arcs":[[4,0,10],[4,3,3],[4,2,3],[4,1,3],[3,0,7],[3,1,3],[2,0,7],[2,1,3],[1,0,5]],"graph":{"inner":{"edge_property":"directed","edges":[[0,1,null],[1,2,null],[2,3,null],[3,4,null]],"node_holes":[],"nodes":[null,null,null,null,null]}}},"samples":[{"config":[0,0,0,1,0,0,0,0,1],"metric":true},{"config":[0,0,0,0,0,0,0,0,0],"metric":false}],"optimal":[{"config":[0,0,0,1,0,0,0,0,1],"metric":true}]},
+    {"problem":"SumOfSquaresPartition","variant":{},"instance":{"bound":240,"num_groups":3,"sizes":[5,3,8,2,7,1]},"samples":[{"config":[1,2,0,1,2,0],"metric":true}],"optimal":[{"config":[0,0,1,0,2,0],"metric":true},{"config":[0,0,1,0,2,1],"metric":true},{"config":[0,0,1,0,2,2],"metric":true},{"config":[0,0,1,1,2,0],"metric":true},{"config":[0,0,1,1,2,1],"metric":true},{"config":[0,0,1,1,2,2],"metric":true},{"config":[0,0,1,2,2,0],"metric":true},{"config":[0,0,1,2,2,1],"metric":true},{"config":[0,0,1,2,2,2],"metric":true},{"config":[0,0,2,0,1,0],"metric":true},{"config":[0,0,2,0,1,1],"metric":true},{"config":[0,0,2,0,1,2],"metric":true},{"config":[0,0,2,1,1,0],"metric":true},{"config":[0,0,2,1,1,1],"metric":true},{"config":[0,0,2,1,1,2],"metric":true},{"config":[0,0,2,2,1,0],"metric":true},{"config":[0,0,2,2,1,1],"metric":true},{"config":[0,0,2,2,1,2],"metric":true},{"config":[0,1,1,0,2,0],"metric":true},{"config":[0,1,1,0,2,2],"metric":true},{"config":[0,1,1,2,2,0],"metric":true},{"config":[0,1,2,0,1,0],"metric":true},{"config":[0,1,2,0,1,1],"metric":true},{"config":[0,1,2,0,1,2],"metric":true},{"config":[0,1,2,2,1,0],"metric":true},{"config":[0,2,1,0,2,0],"metric":true},{"config":[0,2,1,0,2,1],"metric":true},{"config":[0,2,1,0,2,2],"metric":true},{"config":[0,2,1,1,2,0],"metric":true},{"config":[0,2,2,0,1,0],"metric":true},{"config":[0,2,2,0,1,1],"metric":true},{"config":[0,2,2,1,1,0],"metric":true},{"config":[1,0,0,1,2,1],"metric":true},{"config":[1,0,0,1,2,2],"metric":true},{"config":[1,0,0,2,2,1],"metric":true},{"config":[1,0,2,1,0,0],"metric":true},{"config":[1,0,2,1,0,1],"metric":true},{"config":[1,0,2,1,0,2],"metric":true},{"config":[1,0,2,2,0,1],"metric":true},{"config":[1,1,0,0,2,0],"metric":true},{"config":[1,1,0,0,2,1],"metric":true},{"config":[1,1,0,0,2,2],"metric":true},{"config":[1,1,0,1,2,0],"metric":true},{"config":[1,1,0,1,2,1],"metric":true},{"config":[1,1,0,1,2,2],"metric":true},{"config":[1,1,0,2,2,0],"metric":true},{"config":[1,1,0,2,2,1],"metric":true},{"config":[1,1,0,2,2,2],"metric":true},{"config":[1,1,2,0,0,0],"metric":true},{"config":[1,1,2,0,0,1],"metric":true},{"config":[1,1,2,0,0,2],"metric":true},{"config":[1,1,2,1,0,0],"metric":true},{"config":[1,1,2,1,0,1],"metric":true},{"config":[1,1,2,1,0,2],"metric":true},{"config":[1,1,2,2,0,0],"metric":true},{"config":[1,1,2,2,0,1],"metric":true},{"config":[1,1,2,2,0,2],"metric":true},{"config":[1,2,0,0,2,1],"metric":true},{"config":[1,2,0,1,2,0],"metric":true},{"config":[1,2,0,1,2,1],"metric":true},{"config":[1,2,0,1,2,2],"metric":true},{"config":[1,2,2,0,0,1],"metric":true},{"config":[1,2,2,1,0,0],"metric":true},{"config":[1,2,2,1,0,1],"metric":true},{"config":[2,0,0,1,1,2],"metric":true},{"config":[2,0,0,2,1,1],"metric":true},{"config":[2,0,0,2,1,2],"metric":true},{"config":[2,0,1,1,0,2],"metric":true},{"config":[2,0,1,2,0,0],"metric":true},{"config":[2,0,1,2,0,1],"metric":true},{"config":[2,0,1,2,0,2],"metric":true},{"config":[2,1,0,0,1,2],"metric":true},{"config":[2,1,0,2,1,0],"metric":true},{"config":[2,1,0,2,1,1],"metric":true},{"config":[2,1,0,2,1,2],"metric":true},{"config":[2,1,1,0,0,2],"metric":true},{"config":[2,1,1,2,0,0],"metric":true},{"config":[2,1,1,2,0,2],"metric":true},{"config":[2,2,0,0,1,0],"metric":true},{"config":[2,2,0,0,1,1],"metric":true},{"config":[2,2,0,0,1,2],"metric":true},{"config":[2,2,0,1,1,0],"metric":true},{"config":[2,2,0,1,1,1],"metric":true},{"config":[2,2,0,1,1,2],"metric":true},{"config":[2,2,0,2,1,0],"metric":true},{"config":[2,2,0,2,1,1],"metric":true},{"config":[2,2,0,2,1,2],"metric":true},{"config":[2,2,1,0,0,0],"metric":true},{"config":[2,2,1,0,0,1],"metric":true},{"config":[2,2,1,0,0,2],"metric":true},{"config":[2,2,1,1,0,0],"metric":true},{"config":[2,2,1,1,0,1],"metric":true},{"config":[2,2,1,1,0,2],"metric":true},{"config":[2,2,1,2,0,0],"metric":true},{"config":[2,2,1,2,0,1],"metric":true},{"config":[2,2,1,2,0,2],"metric":true}]},
     {"problem":"TravelingSalesman","variant":{"graph":"SimpleGraph","weight":"i32"},"instance":{"edge_weights":[1,3,2,2,3,1],"graph":{"inner":{"edge_property":"undirected","edges":[[0,1,null],[0,2,null],[0,3,null],[1,2,null],[1,3,null],[2,3,null]],"node_holes":[],"nodes":[null,null,null,null]}}},"samples":[{"config":[1,0,1,1,0,1],"metric":{"Valid":6}}],"optimal":[{"config":[1,0,1,1,0,1],"metric":{"Valid":6}}]},
     {"problem":"UndirectedTwoCommodityIntegralFlow","variant":{},"instance":{"capacities":[1,1,2],"graph":{"inner":{"edge_property":"undirected","edges":[[0,2,null],[1,2,null],[2,3,null]],"node_holes":[],"nodes":[null,null,null,null]}},"requirement_1":1,"requirement_2":1,"sink_1":3,"sink_2":3,"source_1":0,"source_2":1},"samples":[{"config":[1,0,0,0,0,0,1,0,1,0,1,0],"metric":true}],"optimal":[{"config":[0,0,1,0,1,0,0,0,1,0,1,0],"metric":true},{"config":[1,0,0,0,0,0,1,0,1,0,1,0],"metric":true}]}
   ],

src/lib.rs

@@ -62,6 +62,7 @@ pub mod prelude {
         MinimumTardinessSequencing, MultiprocessorScheduling, PaintShop,
         SequencingWithReleaseTimesAndDeadlines, SequencingWithinIntervals,
         ShortestCommonSupersequence, StaffScheduling, StringToStringCorrection, SubsetSum,
+        SumOfSquaresPartition,
     };
     pub use crate::models::set::{
         ComparativeContainment, ConsecutiveSets, ExactCoverBy3Sets, MaximumSetPacking,

src/models/misc/mod.rs

@@ -14,6 +14,7 @@
 //! - [`ShortestCommonSupersequence`]: Find a common supersequence of bounded length
 //! - [`StringToStringCorrection`]: String-to-String Correction (derive target via deletions and swaps)
 //! - [`SubsetSum`]: Find a subset summing to exactly a target value
+//! - [`SumOfSquaresPartition`]: Partition integers into K groups minimizing sum of squared group sums
 
 mod bin_packing;
 pub(crate) mod factoring;
@@ -29,6 +30,7 @@ pub(crate) mod shortest_common_supersequence;
 mod staff_scheduling;
 pub(crate) mod string_to_string_correction;
 mod subset_sum;
+pub(crate) mod sum_of_squares_partition;
 
 pub use bin_packing::BinPacking;
 pub use factoring::Factoring;
@@ -44,6 +46,7 @@ pub use shortest_common_supersequence::ShortestCommonSupersequence;
 pub use staff_scheduling::StaffScheduling;
 pub use string_to_string_correction::StringToStringCorrection;
 pub use subset_sum::SubsetSum;
+pub use sum_of_squares_partition::SumOfSquaresPartition;
 
 #[cfg(feature = "example-db")]
 pub(crate) fn canonical_model_example_specs() -> Vec<crate::example_db::specs::ModelExampleSpec> {
@@ -57,6 +60,7 @@ pub(crate) fn canonical_model_example_specs() -> Vec<crate::example_db::specs::M
     specs.extend(shortest_common_supersequence::canonical_model_example_specs());
     specs.extend(string_to_string_correction::canonical_model_example_specs());
     specs.extend(minimum_tardiness_sequencing::canonical_model_example_specs());
+    specs.extend(sum_of_squares_partition::canonical_model_example_specs());
     specs.extend(sequencing_with_release_times_and_deadlines::canonical_model_example_specs());
     specs
 }