feat: add SchedulingToMinimizeWeightedCompletionTime model + ILP rule (#809)

* feat: add SchedulingToMinimizeWeightedCompletionTime model and ILP rule (#505)

Implement the multiprocessor scheduling optimization problem (SS13 from
Garey & Johnson) that minimizes total weighted completion time with
Smith's rule ordering on each processor, plus its direct ILP reduction.

- Model: Min<u64> value type, dims=[m; n], Smith's rule for per-processor ordering
- ILP rule: binary assignment + completion time + ordering variables with big-M constraints
- 16 model tests + 6 ILP tests, all passing
- CLI: pred create support, schema registration
- Paper: problem-def + reduction-rule entries with bibliography

Closes #505

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

* Fix paper definition: replace undefined σ(t) with explicit completion time description

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

* Add C_t <= M upper bounds to ILP reduction per issue #783 spec

Adds completion-time upper bound constraints (C_t <= M where M = Σ l(t))
to tighten the LP relaxation. Updates overhead formula from 2*n to 3*n
bound constraints, matching the 75-constraint canonical example.

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

---------

Co-authored-by: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

Author: isPANN

docs/paper/reductions.typ

@@ -183,6 +183,7 @@
   "RectilinearPictureCompression": [Rectilinear Picture Compression],
   "ResourceConstrainedScheduling": [Resource Constrained Scheduling],
   "RootedTreeStorageAssignment": [Rooted Tree Storage Assignment],
+  "SchedulingToMinimizeWeightedCompletionTime": [Scheduling to Minimize Weighted Completion Time],
   "SchedulingWithIndividualDeadlines": [Scheduling With Individual Deadlines],
   "SequencingToMinimizeMaximumCumulativeCost": [Sequencing to Minimize Maximum Cumulative Cost],
   "SequencingToMinimizeWeightedCompletionTime": [Sequencing to Minimize Weighted Completion Time],
@@ -5743,6 +5744,104 @@ A classical NP-complete problem from Garey and Johnson @garey1979[Ch.~3, p.~76],
     ]
   ]
 }
+#{
+  let x = load-model-example("SchedulingToMinimizeWeightedCompletionTime")
+  let ntasks = x.instance.lengths.len()
+  let m = x.instance.num_processors
+  let lengths = x.instance.lengths
+  let weights = x.instance.weights
+  let sigma = x.optimal_config
+  // Group tasks by processor
+  let tasks-by-proc = range(m).map(p =>
+    range(ntasks).filter(i => sigma.at(i) == p)
+  )
+  [
+    #problem-def("SchedulingToMinimizeWeightedCompletionTime")[
+      Given a finite set $T$ of tasks with processing lengths $ell: T -> ZZ^+$ and weights $w: T -> ZZ^+$, and a number $m in ZZ^+$ of identical processors, find an assignment $p: T -> {1, dots, m}$ that minimizes the total weighted completion time $sum_(t in T) w(t) dot C(t)$, where on each processor tasks are ordered by Smith's rule (non-decreasing $ell(t) "/" w(t)$ ratio) and $C(t)$ is the completion time of task $t$ (i.e., the cumulative processing time up to and including $t$ on its assigned processor).
+    ][
+      Scheduling to Minimize Weighted Completion Time is problem A5 SS13 in Garey & Johnson @garey1979. NP-complete for $m = 2$ by reduction from Partition @lenstra1977, and NP-complete in the strong sense for arbitrary $m$. For a fixed assignment of tasks to processors, Smith's rule gives the optimal ordering on each processor, reducing the search space to $m^n$ processor assignments @smith1956. The problem is solvable in polynomial time when all lengths are equal or when all weights are equal @conway1967 @horn1973.
+
+      *Example.* Let $T = {t_1, dots, t_#ntasks}$ with lengths $(#lengths.map(str).join(", "))$, weights $(#weights.map(str).join(", "))$, and $m = #m$ processors. The optimal assignment $(#sigma.map(v => str(v + 1)).join(", "))$ achieves total weighted completion time #x.optimal_value:
+      #for p in range(m) [
+        - Processor #(p + 1): ${#tasks-by-proc.at(p).map(i => $t_#(i + 1)$).join(", ")}$#if tasks-by-proc.at(p).len() > 0 {
+          let proc-tasks = tasks-by-proc.at(p)
+          let elapsed = 0
+          let contributions = ()
+          for t in proc-tasks {
+            elapsed = elapsed + lengths.at(t)
+            contributions.push($#elapsed times #(weights.at(t)) = #(elapsed * weights.at(t))$)
+          }
+          [ -- contributions: #contributions.join(", ")]
+        }
+      ]
+
+      #pred-commands(
+        "pred create --example " + problem-spec(x) + " -o scheduling-wct.json",
+        "pred solve scheduling-wct.json --solver brute-force",
+        "pred evaluate scheduling-wct.json --config " + x.optimal_config.map(str).join(","),
+      )
+
+      #figure({
+        canvas(length: 1cm, {
+          import draw: *
+          let scale = 0.2
+          let width = 1.2
+          let gap = 0.8
+          let colors = (
+            rgb("#4e79a7"),
+            rgb("#e15759"),
+            rgb("#76b7b2"),
+            rgb("#f28e2b"),
+            rgb("#59a14f"),
+          )
+
+          for p in range(m) {
+            let x0 = p * (width + gap)
+            let max-time = tasks-by-proc.at(p).fold(0, (acc, t) => acc + lengths.at(t))
+            rect((x0, 0), (x0 + width, max-time * scale), stroke: 0.8pt + black)
+            let y = 0
+            for task in tasks-by-proc.at(p) {
+              let len = lengths.at(task)
+              let col = colors.at(task)
+              rect(
+                (x0, y),
+                (x0 + width, y + len * scale),
+                fill: col.transparentize(25%),
+                stroke: 0.4pt + col,
+              )
+              content(
+                (x0 + width / 2, y + len * scale / 2),
+                text(7pt, fill: white)[$t_#(task + 1)$],
+              )
+              y += len * scale
+            }
+            content((x0 + width / 2, -0.3), text(8pt)[$P_#(p + 1)$])
+          }
+        })
+      },
+      caption: [Canonical Scheduling to Minimize Weighted Completion Time instance with #ntasks tasks on #m processors. Tasks are ordered on each processor by Smith's rule.],
+      ) <fig:scheduling-wct>
+    ]
+  ]
+}
+
+// Reduction: SchedulingToMinimizeWeightedCompletionTime -> ILP
+#reduction-rule("SchedulingToMinimizeWeightedCompletionTime", "ILP",
+  example: false,
+)[
+  This $O(n^2 m)$ reduction constructs an ILP with binary assignment variables $x_(t,p)$, integer completion-time variables $C_t$, and binary ordering variables $y_(i,j)$ for task pairs. Big-M disjunctive constraints enforce non-overlapping execution on shared processors.
+][
+  _Construction._ Let $n = |T|$ and $m$ be the number of processors. Create $n m$ binary assignment variables $x_(t,p) in {0, 1}$ (task $t$ on processor $p$), $n$ integer completion-time variables $C_t$, and $n(n-1)/2$ binary ordering variables $y_(i,j)$ for $i < j$. The constraints are:
+  (1) Assignment: $sum_p x_(t,p) = 1$ for each $t$.
+  (2) Completion bounds: $C_t >= ell(t)$ for each $t$.
+  (3) Disjunctive: for each pair $(i,j)$ with $i < j$ and each processor $p$, big-M constraints ensure that if both tasks are on processor $p$, one must complete before the other starts.
+  The objective minimizes $sum_t w(t) dot C_t$.
+
+  _Correctness._ ($arrow.r.double$) Any valid schedule gives a feasible ILP solution with the same objective. ($arrow.l.double$) Any ILP solution encodes a valid assignment and non-overlapping schedule.
+
+  _Solution extraction._ For each task $t$, find the processor $p$ with $x_(t,p) = 1$.
+]
+
 #{
   let x = load-model-example("SequencingWithinIntervals")
   let ntasks = x.instance.lengths.len()

docs/paper/references.bib

@@ -1447,6 +1447,33 @@ @article{edmondsjohnson1973
   year    = {1973}
 }
 
+@article{lenstra1977,
+  author  = {J. K. Lenstra and A. H. G. Rinnooy Kan and P. Brucker},
+  title   = {Complexity of Machine Scheduling Problems},
+  journal = {Annals of Discrete Mathematics},
+  volume  = {1},
+  pages   = {343--362},
+  year    = {1977},
+  doi     = {10.1016/S0167-5060(08)70743-X}
+}
+
+@book{conway1967,
+  author    = {Richard W. Conway and William L. Maxwell and Louis W. Miller},
+  title     = {Theory of Scheduling},
+  publisher = {Addison-Wesley},
+  year      = {1967}
+}
+
+@article{horn1973,
+  author  = {W. A. Horn},
+  title   = {Minimizing Average Flow Time with Parallel Machines},
+  journal = {Operations Research},
+  volume  = {21},
+  number  = {3},
+  pages   = {846--847},
+  year    = {1973}
+}
+
 @techreport{plaisted1976,
   author      = {David A. Plaisted},
   title       = {Some Polynomial and Integer Divisibility Problems Are {NP}-Hard},

problemreductions-cli/src/cli.rs

@@ -280,6 +280,7 @@ Flags by problem type:
   AcyclicPartition                --arcs [--weights] [--arc-costs] --weight-bound --cost-bound [--num-vertices]
   CVP                             --basis, --target-vec [--bounds]
   MultiprocessorScheduling        --lengths, --num-processors, --deadline
+  SchedulingToMinimizeWeightedCompletionTime  --lengths, --weights, --num-processors
   SequencingWithinIntervals       --release-times, --deadlines, --lengths
   OptimalLinearArrangement        --graph
   RootedTreeArrangement           --graph, --bound
@@ -345,6 +346,7 @@ 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 MultiprocessorScheduling --lengths 4,5,3,2,6 --num-processors 2 --deadline 10
+  pred create SchedulingToMinimizeWeightedCompletionTime --lengths 1,2,3,4,5 --weights 6,4,3,2,1 --num-processors 2
   pred create UndirectedFlowLowerBounds --graph 0-1,0-2,1-3,2-3,1-4,3-5,4-5 --capacities 2,2,2,2,1,3,2 --lower-bounds 1,1,0,0,1,0,1 --source 0 --sink 5 --requirement 3
   pred create ConsistencyOfDatabaseFrequencyTables --num-objects 6 --attribute-domains \"2,3,2\" --frequency-tables \"0,1:1,1,1|1,1,1;1,2:1,1|0,2|1,1\" --known-values \"0,0,0;3,0,1;1,2,1\"
   pred create BiconnectivityAugmentation --graph 0-1,1-2,2-3 --potential-edges 0-2:3,0-3:4,1-3:2 --budget 5
@@ -668,7 +670,7 @@ pub struct CreateArgs {
     /// Deadline for FlowShopScheduling, MultiprocessorScheduling, or ResourceConstrainedScheduling
     #[arg(long)]
     pub deadline: Option<u64>,
-    /// Number of processors/machines for FlowShopScheduling, JobShopScheduling, MultiprocessorScheduling, ResourceConstrainedScheduling, or SchedulingWithIndividualDeadlines
+    /// Number of processors/machines for FlowShopScheduling, JobShopScheduling, MultiprocessorScheduling, ResourceConstrainedScheduling, SchedulingToMinimizeWeightedCompletionTime, or SchedulingWithIndividualDeadlines
     #[arg(long)]
     pub num_processors: Option<usize>,
     /// Binary schedule patterns for StaffScheduling (semicolon-separated rows, e.g., "1,1,0;0,1,1")
@@ -919,7 +921,7 @@ mod tests {
         ));
         assert!(
             help.contains(
-                "Number of processors/machines for FlowShopScheduling, JobShopScheduling, MultiprocessorScheduling, ResourceConstrainedScheduling, or SchedulingWithIndividualDeadlines"
+                "Number of processors/machines for FlowShopScheduling, JobShopScheduling, MultiprocessorScheduling, ResourceConstrainedScheduling, SchedulingToMinimizeWeightedCompletionTime, or SchedulingWithIndividualDeadlines"
             ),
             "create help should describe --num-processors for both scheduling models"
         );

problemreductions-cli/src/commands/create.rs

@@ -27,10 +27,11 @@ use problemreductions::models::misc::{
     JobShopScheduling, KnownValue, KthLargestMTuple, LongestCommonSubsequence,
     MinimumTardinessSequencing, MultiprocessorScheduling, PaintShop, PartiallyOrderedKnapsack,
     ProductionPlanning, QueryArg, RectilinearPictureCompression, ResourceConstrainedScheduling,
-    SchedulingWithIndividualDeadlines, SequencingToMinimizeMaximumCumulativeCost,
-    SequencingToMinimizeWeightedCompletionTime, SequencingToMinimizeWeightedTardiness,
-    SequencingWithReleaseTimesAndDeadlines, SequencingWithinIntervals, ShortestCommonSupersequence,
-    StringToStringCorrection, SubsetSum, SumOfSquaresPartition, ThreePartition, TimetableDesign,
+    SchedulingToMinimizeWeightedCompletionTime, SchedulingWithIndividualDeadlines,
+    SequencingToMinimizeMaximumCumulativeCost, SequencingToMinimizeWeightedCompletionTime,
+    SequencingToMinimizeWeightedTardiness, SequencingWithReleaseTimesAndDeadlines,
+    SequencingWithinIntervals, ShortestCommonSupersequence, StringToStringCorrection, SubsetSum,
+    SumOfSquaresPartition, ThreePartition, TimetableDesign,
 };
 use problemreductions::models::BiconnectivityAugmentation;
 use problemreductions::prelude::*;
@@ -677,6 +678,9 @@ fn example_for(canonical: &str, graph_type: Option<&str>) -> &'static str {
             "--num-periods 6 --demands 5,3,7,2,8,5 --capacities 12,12,12,12,12,12 --setup-costs 10,10,10,10,10,10 --production-costs 1,1,1,1,1,1 --inventory-costs 1,1,1,1,1,1 --cost-bound 80"
         }
         "MultiprocessorScheduling" => "--lengths 4,5,3,2,6 --num-processors 2 --deadline 10",
+        "SchedulingToMinimizeWeightedCompletionTime" => {
+            "--lengths 1,2,3,4,5 --weights 6,4,3,2,1 --num-processors 2"
+        }
         "JobShopScheduling" => {
             "--job-tasks \"0:3,1:4;1:2,0:3,1:2;0:4,1:3;1:5,0:2;0:2,1:3,0:1\" --num-processors 2"
         }
@@ -3283,6 +3287,37 @@ pub fn create(args: &CreateArgs, out: &OutputConfig) -> Result<()> {
             )
         }
 
+        // SchedulingToMinimizeWeightedCompletionTime
+        "SchedulingToMinimizeWeightedCompletionTime" => {
+            let usage = "Usage: pred create SchedulingToMinimizeWeightedCompletionTime --lengths 1,2,3,4,5 --weights 6,4,3,2,1 --num-processors 2";
+            let lengths_str = args.lengths.as_deref().ok_or_else(|| {
+                anyhow::anyhow!(
+                    "SchedulingToMinimizeWeightedCompletionTime requires --lengths, --weights, and --num-processors\n\n{usage}"
+                )
+            })?;
+            let weights_str = args.weights.as_deref().ok_or_else(|| {
+                anyhow::anyhow!(
+                    "SchedulingToMinimizeWeightedCompletionTime requires --weights\n\n{usage}"
+                )
+            })?;
+            let num_processors = args.num_processors.ok_or_else(|| {
+                anyhow::anyhow!("SchedulingToMinimizeWeightedCompletionTime requires --num-processors\n\n{usage}")
+            })?;
+            if num_processors == 0 {
+                bail!("SchedulingToMinimizeWeightedCompletionTime requires --num-processors > 0\n\n{usage}");
+            }
+            let lengths: Vec<u64> = util::parse_comma_list(lengths_str)?;
+            let weights: Vec<u64> = util::parse_comma_list(weights_str)?;
+            (
+                ser(SchedulingToMinimizeWeightedCompletionTime::new(
+                    lengths,
+                    weights,
+                    num_processors,
+                ))?,
+                resolved_variant.clone(),
+            )
+        }
+
         "CapacityAssignment" => {
             let usage = "Usage: pred create CapacityAssignment --capacities 1,2,3 --cost-matrix \"1,3,6;2,4,7;1,2,5\" --delay-matrix \"8,4,1;7,3,1;6,3,1\" --delay-budget 12";
             let capacities_str = args.capacities.as_deref().ok_or_else(|| {

src/models/misc/mod.rs

@@ -88,6 +88,7 @@ mod precedence_constrained_scheduling;
 mod production_planning;
 mod rectilinear_picture_compression;
 pub(crate) mod resource_constrained_scheduling;
+mod scheduling_to_minimize_weighted_completion_time;
 mod scheduling_with_individual_deadlines;
 mod sequencing_to_minimize_maximum_cumulative_cost;
 mod sequencing_to_minimize_weighted_completion_time;
@@ -131,6 +132,7 @@ pub use precedence_constrained_scheduling::PrecedenceConstrainedScheduling;
 pub use production_planning::ProductionPlanning;
 pub use rectilinear_picture_compression::RectilinearPictureCompression;
 pub use resource_constrained_scheduling::ResourceConstrainedScheduling;
+pub use scheduling_to_minimize_weighted_completion_time::SchedulingToMinimizeWeightedCompletionTime;
 pub use scheduling_with_individual_deadlines::SchedulingWithIndividualDeadlines;
 pub use sequencing_to_minimize_maximum_cumulative_cost::SequencingToMinimizeMaximumCumulativeCost;
 pub use sequencing_to_minimize_weighted_completion_time::SequencingToMinimizeWeightedCompletionTime;
@@ -164,6 +166,7 @@ pub(crate) fn canonical_model_example_specs() -> Vec<crate::example_db::specs::M
     specs.extend(partition::canonical_model_example_specs());
     specs.extend(production_planning::canonical_model_example_specs());
     specs.extend(rectilinear_picture_compression::canonical_model_example_specs());
+    specs.extend(scheduling_to_minimize_weighted_completion_time::canonical_model_example_specs());
     specs.extend(scheduling_with_individual_deadlines::canonical_model_example_specs());
     specs.extend(sequencing_within_intervals::canonical_model_example_specs());
     specs.extend(staff_scheduling::canonical_model_example_specs());