Fix #212: [Model] MultiprocessorScheduling

docs/paper/reductions.typ

@@ -114,6 +114,7 @@
   "SubgraphIsomorphism": [Subgraph Isomorphism],
   "PartitionIntoTriangles": [Partition Into Triangles],
   "FlowShopScheduling": [Flow Shop Scheduling],
+  "MultiprocessorScheduling": [Multiprocessor Scheduling],
   "MinimumTardinessSequencing": [Minimum Tardiness Sequencing],
   "SequencingWithinIntervals": [Sequencing Within Intervals],
   "DirectedTwoCommodityIntegralFlow": [Directed Two-Commodity Integral Flow],
@@ -2377,6 +2378,75 @@ NP-completeness was established by Garey, Johnson, and Stockmeyer @gareyJohnsonS
   ) <fig:flowshop>
 ]
 
+#{
+  let x = load-model-example("MultiprocessorScheduling")
+  let lengths = x.instance.lengths
+  let num-processors = x.instance.num_processors
+  let deadline = x.instance.deadline
+  let assignment = x.optimal.at(0).config
+  let tasks-by-processor = range(num-processors).map(p =>
+    range(lengths.len()).filter(i => assignment.at(i) == p)
+  )
+  let loads = tasks-by-processor.map(tasks => tasks.map(i => lengths.at(i)).sum())
+  let max-x = (num-processors - 1) * 1.8 + 1.0
+  [
+    #problem-def("MultiprocessorScheduling")[
+      Given a finite set $T$ of tasks with processing lengths $ell: T -> ZZ^+$, a number $m in ZZ^+$ of identical processors, and a deadline $D in ZZ^+$, determine whether there exists an assignment $p: T -> {1, dots, m}$ such that for every processor $i in {1, dots, m}$ we have $sum_(t in T: p(t) = i) ell(t) <= D$.
+    ][
+      Multiprocessor Scheduling is problem SS8 in Garey & Johnson @garey1979. Their original formulation uses start times on identical processors, but because tasks are independent and non-preemptive, any feasible schedule can be packed contiguously on each processor. The model implemented here therefore uses processor-assignment variables, and feasibility reduces to checking that every processor's total load is at most $D$. For fixed $m$, dynamic programming over load vectors gives pseudo-polynomial algorithms; for general $m$, the best known exact algorithm runs in $O^*(2^n)$ time via inclusion-exclusion over set partitions @bjorklund2009.
+
+      *Example.* Let $T = {t_1, dots, t_5}$ with lengths $(4, 5, 3, 2, 6)$, $m = 2$, and $D = 10$. The satisfying assignment $(1, 2, 2, 2, 1)$ places $t_1$ and $t_5$ on processor 1 and $t_2, t_3, t_4$ on processor 2. The verifier computes the processor loads $4 + 6 = 10$ and $5 + 3 + 2 = 10$, so both meet the deadline exactly.
+
+      #figure({
+        canvas(length: 1cm, {
+          let scale = 0.25
+          let width = 1.0
+          let gap = 0.8
+          let colors = (
+            rgb("#4e79a7"),
+            rgb("#e15759"),
+            rgb("#76b7b2"),
+            rgb("#f28e2b"),
+            rgb("#59a14f"),
+          )
+
+          for p in range(num-processors) {
+            let x0 = p * (width + gap)
+            draw.rect((x0, 0), (x0 + width, deadline * scale), stroke: 0.8pt + black)
+            let y = 0
+            for task in tasks-by-processor.at(p) {
+              let len = lengths.at(task)
+              let col = colors.at(task)
+              draw.rect(
+                (x0, y),
+                (x0 + width, y + len * scale),
+                fill: col.transparentize(25%),
+                stroke: 0.4pt + col,
+              )
+              draw.content(
+                (x0 + width / 2, y + len * scale / 2),
+                text(7pt, fill: white)[$t_#(task + 1)$],
+              )
+              y += len * scale
+            }
+            draw.content((x0 + width / 2, -0.3), text(8pt)[$P_#(p + 1)$])
+            draw.content((x0 + width / 2, deadline * scale + 0.25), text(7pt)[$L_#(p + 1) = #loads.at(p)$])
+          }
+
+          draw.line(
+            (-0.15, deadline * scale),
+            (max-x + 0.15, deadline * scale),
+            stroke: (dash: "dashed", paint: luma(150), thickness: 0.5pt),
+          )
+          draw.content((-0.45, deadline * scale), text(7pt)[$D$])
+        })
+      },
+      caption: [Canonical Multiprocessor Scheduling instance with 5 tasks on 2 processors. Stacked blocks show the satisfying assignment $(1, 2, 2, 2, 1)$; both processor loads equal the deadline $D = 10$.],
+      ) <fig:multiprocessor-scheduling>
+    ]
+  ]
+}
+
 #{
   let x = load-model-example("SequencingWithinIntervals")
   let ntasks = x.instance.lengths.len()

docs/paper/references.bib

@@ -40,6 +40,16 @@ @article{shang2018
   doi     = {10.1016/j.cor.2017.10.015}
 }
 
+@article{brucker1977,
+  author  = {Peter Brucker and Jan Karel Lenstra and Alexander H. G. Rinnooy Kan},
+  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}
+}
+
 @inproceedings{karp1972,
   author    = {Richard M. Karp},
   title     = {Reducibility among Combinatorial Problems},

docs/src/cli.md

@@ -515,8 +515,9 @@ Source evaluation: Valid(2)
 ```
 
 > **Note:** The ILP solver requires a reduction path from the target problem to ILP.
-> Some problems (e.g., BoundedComponentSpanningForest, LengthBoundedDisjointPaths) do not currently have one, so use
-> `pred solve <file> --solver brute-force` for these.
+> Some problems do not currently have one. Examples include BoundedComponentSpanningForest,
+> LengthBoundedDisjointPaths, QUBO, SpinGlass, MaxCut, CircuitSAT, and MultiprocessorScheduling.
+> Use `pred solve <file> --solver brute-force` for these, or reduce to a problem that supports ILP first.
 > For other problems, use `pred path <PROBLEM> ILP` to check whether an ILP reduction path exists.
 
 ## Shell Completions

problemreductions-cli/src/cli.rs

@@ -243,6 +243,7 @@ Flags by problem type:
   BMF                             --matrix (0/1), --rank
   SteinerTree                     --graph, --edge-weights, --terminals
   CVP                             --basis, --target-vec [--bounds]
+  MultiprocessorScheduling        --lengths, --num-processors, --deadline
   SequencingWithinIntervals       --release-times, --deadlines, --lengths
   OptimalLinearArrangement        --graph, --bound
   RuralPostman (RPP)              --graph, --edge-weights, --required-edges, --bound
@@ -276,6 +277,7 @@ Examples:
   pred create MIS/KingsSubgraph --positions \"0,0;1,0;1,1;0,1\"
   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 BiconnectivityAugmentation --graph 0-1,1-2,2-3 --potential-edges 0-2:3,0-3:4,1-3:2 --budget 5
   pred create FVS --arcs \"0>1,1>2,2>0\" --weights 1,1,1
   pred create UndirectedTwoCommodityIntegralFlow --graph 0-2,1-2,2-3 --capacities 1,1,2 --source-1 0 --sink-1 3 --source-2 1 --sink-2 3 --requirement-1 1 --requirement-2 1
@@ -432,7 +434,7 @@ pub struct CreateArgs {
     /// Release times for SequencingWithinIntervals (comma-separated, e.g., "0,0,5")
     #[arg(long)]
     pub release_times: Option<String>,
-    /// Processing lengths for SequencingWithinIntervals (comma-separated, e.g., "3,1,1")
+    /// Processing lengths (comma-separated, e.g., "4,5,3,2,6")
     #[arg(long)]
     pub lengths: Option<String>,
     /// Terminal vertices for SteinerTree or MinimumMultiwayCut (comma-separated indices, e.g., "0,2,4")
@@ -474,10 +476,10 @@ pub struct CreateArgs {
     /// Task lengths for FlowShopScheduling (semicolon-separated rows: "3,4,2;2,3,5;4,1,3")
     #[arg(long)]
     pub task_lengths: Option<String>,
-    /// Deadline for FlowShopScheduling
+    /// Deadline for FlowShopScheduling or MultiprocessorScheduling
     #[arg(long)]
     pub deadline: Option<u64>,
-    /// Number of processors/machines for FlowShopScheduling
+    /// Number of processors/machines for FlowShopScheduling or MultiprocessorScheduling
     #[arg(long)]
     pub num_processors: Option<usize>,
     /// Alphabet size for SCS (optional; inferred from max symbol + 1 if omitted)

problemreductions-cli/src/commands/create.rs

@@ -14,7 +14,8 @@ use problemreductions::models::graph::{
 };
 use problemreductions::models::misc::{
     BinPacking, FlowShopScheduling, LongestCommonSubsequence, MinimumTardinessSequencing,
-    PaintShop, SequencingWithinIntervals, ShortestCommonSupersequence, SubsetSum,
+    MultiprocessorScheduling, PaintShop, SequencingWithinIntervals, ShortestCommonSupersequence,
+    SubsetSum,
 };
 use problemreductions::models::BiconnectivityAugmentation;
 use problemreductions::prelude::*;
@@ -226,7 +227,7 @@ fn type_format_hint(type_name: &str, graph_type: Option<&str>) -> &'static str {
             Some("UnitDiskGraph") => "float positions: \"0.0,0.0;1.0,0.0\"",
             _ => "edge list: 0-1,1-2,2-3",
         },
-        "Vec<u64>" => "comma-separated integers: 1,2,3",
+        "Vec<u64>" => "comma-separated integers: 4,5,3,2,6",
         "Vec<W>" => "comma-separated: 1,2,3",
         "Vec<usize>" => "comma-separated indices: 0,2,4",
         "Vec<(usize, usize, W)>" | "Vec<(usize,usize,W)>" => {
@@ -288,6 +289,7 @@ fn example_for(canonical: &str, graph_type: Option<&str>) -> &'static str {
         }
         "PartitionIntoTriangles" => "--graph 0-1,1-2,0-2",
         "Factoring" => "--target 15 --m 4 --n 4",
+        "MultiprocessorScheduling" => "--lengths 4,5,3,2,6 --num-processors 2 --deadline 10",
         "MinimumMultiwayCut" => "--graph 0-1,1-2,2-3 --terminals 0,2 --edge-weights 1,1,1",
         "SequencingWithinIntervals" => "--release-times 0,0,5 --deadlines 11,11,6 --lengths 3,1,1",
         "SteinerTree" => "--graph 0-1,1-2,1-3,3-4 --edge-weights 2,2,1,1 --terminals 0,2,4",
@@ -1286,6 +1288,34 @@ pub fn create(args: &CreateArgs, out: &OutputConfig) -> Result<()> {
             )
         }
 
+        // MultiprocessorScheduling
+        "MultiprocessorScheduling" => {
+            let usage = "Usage: pred create MultiprocessorScheduling --lengths 4,5,3,2,6 --num-processors 2 --deadline 10";
+            let lengths_str = args.lengths.as_deref().ok_or_else(|| {
+                anyhow::anyhow!(
+                    "MultiprocessorScheduling requires --lengths, --num-processors, and --deadline\n\n{usage}"
+                )
+            })?;
+            let num_processors = args.num_processors.ok_or_else(|| {
+                anyhow::anyhow!("MultiprocessorScheduling requires --num-processors\n\n{usage}")
+            })?;
+            if num_processors == 0 {
+                bail!("MultiprocessorScheduling requires --num-processors > 0\n\n{usage}");
+            }
+            let deadline = args.deadline.ok_or_else(|| {
+                anyhow::anyhow!("MultiprocessorScheduling requires --deadline\n\n{usage}")
+            })?;
+            let lengths: Vec<u64> = util::parse_comma_list(lengths_str)?;
+            (
+                ser(MultiprocessorScheduling::new(
+                    lengths,
+                    num_processors,
+                    deadline,
+                ))?,
+                resolved_variant.clone(),
+            )
+        }
+
         // MinimumMultiwayCut
         "MinimumMultiwayCut" => {
             let (graph, _) = parse_graph(args).map_err(|e| {
@@ -1708,7 +1738,6 @@ pub fn create(args: &CreateArgs, out: &OutputConfig) -> Result<()> {
                 resolved_variant.clone(),
             )
         }
-
         _ => bail!("{}", crate::problem_name::unknown_problem_error(canonical)),
     };
 

problemreductions-cli/src/commands/inspect.rs

@@ -40,8 +40,17 @@ fn inspect_problem(pj: &ProblemJson, out: &OutputConfig) -> Result<()> {
     }
     text.push_str(&format!("Variables: {}\n", problem.num_variables_dyn()));
 
-    // Solvers
-    text.push_str("Solvers: ilp (default), brute-force\n");
+    let solvers = if problem.supports_ilp_solver() {
+        vec!["ilp", "brute-force"]
+    } else {
+        vec!["brute-force"]
+    };
+    let solver_summary = if solvers.first() == Some(&"ilp") {
+        "ilp (default), brute-force".to_string()
+    } else {
+        "brute-force".to_string()
+    };
+    text.push_str(&format!("Solvers: {solver_summary}\n"));
 
     // Reductions
     let outgoing = graph.outgoing_reductions(name);
@@ -56,7 +65,7 @@ fn inspect_problem(pj: &ProblemJson, out: &OutputConfig) -> Result<()> {
         "variant": variant,
         "size_fields": size_fields,
         "num_variables": problem.num_variables_dyn(),
-        "solvers": ["ilp", "brute-force"],
+        "solvers": solvers,
         "reduces_to": targets,
     });
 

problemreductions-cli/src/commands/solve.rs

@@ -97,7 +97,7 @@ fn solve_problem(
             result
         }
         "ilp" => {
-            let result = problem.solve_with_ilp()?;
+            let result = problem.solve_with_ilp().map_err(add_ilp_solver_hint)?;
             let solver_desc = if name == "ILP" {
                 "ilp".to_string()
             } else {
@@ -139,7 +139,7 @@ fn solve_bundle(bundle: ReductionBundle, solver_name: &str, out: &OutputConfig)
     // 2. Solve the target problem
     let target_result = match solver_name {
         "brute-force" => target.solve_brute_force()?,
-        "ilp" => target.solve_with_ilp()?,
+        "ilp" => target.solve_with_ilp().map_err(add_ilp_solver_hint)?,
         _ => unreachable!(),
     };
 
@@ -200,3 +200,14 @@ fn solve_bundle(bundle: ReductionBundle, solver_name: &str, out: &OutputConfig)
     }
     result
 }
+
+fn add_ilp_solver_hint(err: anyhow::Error) -> anyhow::Error {
+    let message = err.to_string();
+    if message.starts_with("No reduction path from ") && message.ends_with(" to ILP") {
+        anyhow::anyhow!(
+            "{message}\n\nHint: try `--solver brute-force` for direct exhaustive search on small instances."
+        )
+    } else {
+        err
+    }
+}

problemreductions-cli/src/dispatch.rs

@@ -47,53 +47,63 @@ impl LoadedProblem {
         Ok(SolveResult { config, evaluation })
     }
 
+    pub fn supports_ilp_solver(&self) -> bool {
+        let name = self.problem_name();
+        name == "ILP" || self.best_ilp_reduction_path().is_some()
+    }
+
     /// Solve using the ILP solver. If the problem is not ILP, auto-reduce to ILP first.
     pub fn solve_with_ilp(&self) -> Result<SolveResult> {
         let name = self.problem_name();
         if name == "ILP" {
             return solve_ilp(self.as_any());
         }
 
-        // Auto-reduce to ILP, solve, and map solution back
+        let reduction_path = self.best_ilp_reduction_path().ok_or_else(|| {
+            anyhow::anyhow!(
+                "No reduction path from {} to ILP. Try `--solver brute-force`, or reduce to a problem that supports ILP.",
+                name
+            )
+        })?;
+        let graph = ReductionGraph::new();
+
+        let chain = graph
+            .reduce_along_path(&reduction_path, self.as_any())
+            .ok_or_else(|| anyhow::anyhow!("Failed to execute reduction chain to ILP"))?;
+
+        let ilp_result = solve_ilp(chain.target_problem_any())?;
+        let config = chain.extract_solution(&ilp_result.config);
+        let evaluation = self.evaluate_dyn(&config);
+        Ok(SolveResult { config, evaluation })
+    }
+
+    fn best_ilp_reduction_path(&self) -> Option<problemreductions::rules::ReductionPath> {
+        let name = self.problem_name();
         let source_variant = self.variant_map();
         let graph = ReductionGraph::new();
         let ilp_variants = graph.variants_for("ILP");
         let input_size = ProblemSize::new(vec![]);
 
         let mut best_path = None;
         for dv in &ilp_variants {
-            if let Some(p) = graph.find_cheapest_path(
+            if let Some(path) = graph.find_cheapest_path(
                 name,
                 &source_variant,
                 "ILP",
                 dv,
                 &input_size,
                 &MinimizeSteps,
             ) {
-                let is_better = best_path
-                    .as_ref()
-                    .is_none_or(|bp: &problemreductions::rules::ReductionPath| p.len() < bp.len());
+                let is_better = best_path.as_ref().is_none_or(
+                    |current: &problemreductions::rules::ReductionPath| path.len() < current.len(),
+                );
                 if is_better {
-                    best_path = Some(p);
+                    best_path = Some(path);
                 }
             }
         }
 
-        let reduction_path = best_path.ok_or_else(|| {
-            anyhow::anyhow!(
-                "No reduction path from {} to ILP. Try `--solver brute-force`, or reduce to a problem that supports ILP.",
-                name
-            )
-        })?;
-
-        let chain = graph
-            .reduce_along_path(&reduction_path, self.as_any())
-            .ok_or_else(|| anyhow::anyhow!("Failed to execute reduction chain to ILP"))?;
-
-        let ilp_result = solve_ilp(chain.target_problem_any())?;
-        let config = chain.extract_solution(&ilp_result.config);
-        let evaluation = self.evaluate_dyn(&config);
-        Ok(SolveResult { config, evaluation })
+        best_path
     }
 }
 
@@ -249,4 +259,26 @@ mod tests {
         let json = serialize_any_problem("BinPacking", &variant, &problem as &dyn Any).unwrap();
         assert_eq!(json, serde_json::to_value(&problem).unwrap());
     }
+
+    #[test]
+    fn test_load_problem_rejects_zero_processor_multiprocessor_scheduling() {
+        let loaded = load_problem(
+            "MultiprocessorScheduling",
+            &BTreeMap::new(),
+            serde_json::json!({
+                "lengths": [1, 2],
+                "num_processors": 0,
+                "deadline": 5
+            }),
+        );
+        assert!(
+            loaded.is_err(),
+            "zero-processor instance should be rejected"
+        );
+        let err = loaded.err().unwrap();
+        assert!(
+            err.to_string().contains("expected positive integer, got 0"),
+            "unexpected error: {err}"
+        );
+    }
 }