[Model] OpenShopScheduling

[isPANN]

Important

Build as optimization, not decision. Value = Min<usize>.
Objective: Minimize makespan (completion time of the last task).
Do not add a bound/threshold field — let the solver find the optimum directly. See #765.

Motivation

OPEN-SHOP SCHEDULING (P198) from Garey & Johnson, A5 SS14. A classical NP-hard scheduling problem in which each job consists of one task per machine, and tasks of the same job may be processed in any order (unlike flow-shop where the order is fixed, or job-shop where each job has a prescribed route). The goal is to minimize the makespan. NP-complete for m ≥ 3 machines by reduction from PARTITION (Gonzalez & Sahni, 1976). Polynomial for m = 2 machines and for the preemptive variant with any number of machines.

Associated rules:

• R143: Partition → OpenShopScheduling (this model is the target)

Definition

Name: OpenShopScheduling
Reference: Garey & Johnson, Computers and Intractability, A5 SS14

Mathematical definition:

INSTANCE: Number m ∈ Z⁺ of machines, set J of n jobs, each job j ∈ J consisting of m tasks t₁[j], t₂[j], …, tₘ[j] (task tᵢ[j] is executed by machine i), and for each task a processing time p(j, i) ∈ Z⁰⁺.

OBJECTIVE: Find an open-shop schedule — a collection of one-machine schedules σᵢ for i = 1, …, m — that minimizes the makespan (completion time of the last task), subject to:

1. Machine constraint: On each machine i, tasks do not overlap (non-preemptive).
2. Job constraint: For each job j, its tasks on different machines do not overlap in time (a job occupies at most one machine at a time).

The corresponding decision problem (GJ SS14) asks whether a schedule exists with makespan ≤ D for a given deadline D.

Variables

• Count: n × m (one variable per job-machine pair, representing the start time). The start time domain is bounded by D_max = Σ_j max_i p(j,i) · n (a loose upper bound).
• Per-variable domain: For ILP formulation, binary ordering variables x_{j,k,i} (job j before job k on machine i) plus integer start time variables. For brute-force, enumerate orderings of jobs on each machine: (n!)^m permutations.
• Meaning: σᵢ(j) is the start time of job j's task on machine i. The evaluate function computes the makespan and checks both constraints.

dims() returns [n; n × m] (permutation of n jobs on each of m machines).

Schema (data type)

Type name: OpenShopScheduling
Variants: none (no type parameters)

Field	Type	Description	Getter
num_machines	usize	Number of machines m	num_machines()
processing_times	Vec<Vec<usize>>	Processing time matrix: p[j][i] for job j, machine i (n × m)	num_jobs() → self.processing_times.len()

Problem type: Optimization (minimization)
Value type: Min<usize>

Complexity

• Decision complexity: NP-complete for m ≥ 3 (Gonzalez & Sahni, 1976; transformation from PARTITION). NP-complete in the strong sense for arbitrary m (Lenstra, 1977). Polynomial for m = 2.
• Best known exact algorithm: Brute-force enumerates (n!)^m orderings across all machines. For the general case, exact methods use ILP or constraint programming.
• Complexity string for declare_variants!: "factorial(num_jobs)^num_machines" (enumerate all orderings of jobs on each machine)
• Special cases:
  • m = 2: polynomial (Gonzalez & Sahni, 1976)
  • Preemptive variant: polynomial for any m (Gonzalez & Sahni, 1976)
  • Preemptive with release times: polynomial for m = 2 (Cho & Sahni, 1978)
• References:
  • [Gonzalez & Sahni, 1976] T. Gonzalez and S. Sahni, "Open shop scheduling to minimize finish time", JACM 23(4), pp. 665-679, 1976.
  • [Lenstra, 1977] J. K. Lenstra, "Sequencing by enumerative methods", Mathematisch Centrum, Amsterdam, 1977.
  • [Cho & Sahni, 1978] Y. Cho and S. Sahni, "Preemptive scheduling of independent jobs with release and due dates on open, flow and job shop", 1978.

Extra Remark

Full book text:

INSTANCE: Number m in Z+ of processors, set J of jobs, each job j in J consisting of m tasks t1[j],t2[j], ..., tm[j] (with ti[j] to be executed by processor i), a length l(t) in Z0+ for each such task t, and an overall deadline D in Z+.
QUESTION: Is there an open-shop schedule for J that meets the deadline, i.e., a collection of one-processor schedules sigmai: J->Z0+, 1 <= i <= m, such that sigmai(j) > sigmai(k) implies sigmai(j) >= sigmai(k) + l(ti[k]), such that for each j in J the intervals [sigmai(j), sigmai(j) + l(ti[j])) are all disjoint, and such that sigmai(j) + l(ti[j]) <= D for 1 <= i <= m, 1 <= j <= |J|?

Reference: [Gonzalez and Sahni, 1976]. Transformation from PARTITION.

Comment: Remains NP-complete if m = 3, but can be solved in polynomial time if m = 2. NP-complete in the strong sense for m arbitrary [Lenstra, 1977]. The general problem is solvable in polynomial time if "preemptive" schedules are allowed [Gonzalez and Sahni, 1976], even if two distinct release times are allowed [Cho and Sahni, 1978].

Note: The original GJ formulation is a decision problem with deadline D. This issue reformulates it as an optimization problem (minimize makespan). The decision version is recovered by checking whether the optimal makespan ≤ D.

How to solve

• It can be solved by (existing) bruteforce. Enumerate all (n!)^m orderings of jobs on each machine; for each, greedily compute start times respecting job constraints; return minimum makespan.
• It can be solved by reducing to integer programming. Binary ILP with ordering variables x_{j,k,i} ∈ {0,1} (job j before k on machine i), integer start times σᵢ(j), non-overlap constraints, and makespan minimization.
• Other: Polynomial for m = 2 (Gonzalez-Sahni); preemptive variant solvable in polynomial time by LP.

Example Instance

Input:
m = 3 machines, J = {J₁, J₂, J₃, J₄} (n = 4 jobs)

Processing time matrix p[j][i]:

Job	Machine 1	Machine 2	Machine 3
J₁	3	1	2
J₂	2	3	1
J₃	1	2	3
J₄	2	2	1

Per-machine totals: M₁ = 8, M₂ = 8, M₃ = 7. Per-job totals: J₁ = 6, J₂ = 6, J₃ = 6, J₄ = 5.
Lower bound: max(8, 6) = 8.

Optimal schedule (makespan = 11):

Machine	Schedule
M₁	J₃[0,1), J₂[1,3), J₁[3,6), J₄[6,8)
M₂	J₃[1,3), J₂[3,6), J₁[6,7), J₄[8,10)
M₃	J₃[3,6), J₁[7,9), J₂[9,10), J₄[10,11)

Job timelines (verify no overlap within each job):

• J₁: M₁[3,6) → M₂[6,7) → M₃[7,9) — disjoint ✓
• J₂: M₁[1,3) → M₂[3,6) → M₃[9,10) — disjoint ✓
• J₃: M₁[0,1) → M₂[1,3) → M₃[3,6) — disjoint ✓
• J₄: M₁[6,8) → M₂[8,10) → M₃[10,11) — disjoint ✓

Makespan = 11 (J₄ finishes last on M₃ at time 11).

Suboptimal feasible schedules (confirming non-triviality):

• Makespan 12: 30 valid orderings achieve this
• Makespan 13: 113 valid orderings
• Makespan 15+: majority of the 13,824 orderings

Out of 13,824 total machine-ordering combinations (24³), only 4 achieve the optimal makespan of 11, with 13 distinct makespan values ranging from 11 to 23.

Expected outcome: Min(11)

Reduction Rule Crossref

• R143: Partition → OpenShopScheduling
• Direct ILP: OpenShopScheduling → ILP (ordering + start time formulation)

[zazabap]

Issue Quality Check — Model

Check	Result	Details
Usefulness	✅ Pass	Novel problem not yet in the reduction graph; associated rule R143 (Partition → this) planned
Non-trivial	✅ Pass	Distinct scheduling problem with job×machine processing matrix and open-shop constraints — not isomorphic to any existing model
Correctness	✅ Pass	Gonzalez & Sahni 1976 (JACM 23:665–679) verified; NP-completeness for m≥3 confirmed; polynomial preemptive case confirmed
Well-written	❌ Fail	Example is a sketch (not fully worked); no concrete complexity bound; uniform processing times in example

Overall: 3 passed, 0 warnings, 1 failed

---

Usefulness

OpenShopScheduling does not exist in the current codebase. The problem is a classic NP-complete scheduling problem from Garey & Johnson (A5 SS14) with a planned reduction from Partition (R143).

Non-trivial

This is a genuinely distinct problem. The input structure (jobs × machines processing time matrix with open-shop ordering freedom) differs from all existing models. Open-shop scheduling is not a variant of any existing problem — the constraint that each job has exactly one task per machine, and tasks of the same job can be processed in any order, is unique.

Correctness

All cited references verified:

• Gonzalez & Sahni, 1976: "Open Shop Scheduling to Minimize Finish Time," JACM 23(4):665–679. NP-completeness for m ≥ 3 by reduction from Partition confirmed. Polynomial-time algorithm for preemptive case confirmed.
• Lenstra, 1977: NP-completeness in the strong sense for arbitrary m — consistent with known results.
• Garey & Johnson, 1979: A5 SS14 classification confirmed.

Well-written

Failures:

1. Example is not fully worked. The schedule is described as a "sketch" with vague statements like "Jobs {J_1, J_5} fill gaps on one side" and "Jobs {J_2, J_3, J_4} fill gaps on the other side." A proper example must show the exact start times sigma_i(j) for every job on every machine, and verify all three constraint types (no machine overlap, no job overlap across machines, deadline met). Without explicit start times, the example is not verifiable.

2. No concrete complexity bound. The Complexity section says "exact approaches use branch-and-bound, ILP, or constraint programming" but provides no specific worst-case bound for declare_variants!. A concrete complexity string is required.

3. Uniform processing times in example. Every job has the same processing time on all three machines (J_1: 4/4/4, J_2: 5/5/5, etc.). This is essentially a special case where the open-shop structure collapses — the example should use varied processing times across machines (e.g., J_1 takes 4 on M1, 2 on M2, 5 on M3) to illustrate why open-shop ordering freedom matters.

4. Variable domain overly vague. The domain is stated as {0, 1, ..., D - max_length} but dims() implementation would need a clearer specification. The search space for brute-force is not well-defined — how many possible start times per variable? This affects the dims() implementation.

Recommendations

• Fully work out the example: provide an explicit table of start times for all 6 jobs × 3 machines, then verify each constraint.
• Use non-uniform processing times to make the example illustrative of open-shop flexibility.
• Provide a concrete complexity string (even if just the brute-force bound).
• Clarify the variable encoding for dims() — e.g., each variable is a permutation position or a discretized start time.

[isPANN]

Implementation note: build as optimization

This model should be implemented as an optimization problem (Value = Min<usize> or similar), not a decision problem with a bound field.

• Objective: Minimize makespan (completion time of the last task) across all jobs
• Do not add a deadline D as a threshold — let the solver find the minimum makespan directly
• Same family as JobShopScheduling ([Model] JobShopScheduling #510) and FlowShopScheduling — all should be Min(makespan)

See #765 for context on the decision-vs-optimization policy.

[isPANN]

Issue Quality Check — Model

Check	Result	Details
Usefulness	✅ Pass	Novel problem not yet in reduction graph; rule #481 (Partition -> OpenShopScheduling) planned
Non-trivial	✅ Pass	Distinct from JobShopScheduling and FlowShopScheduling — unique open-shop task ordering freedom
Correctness	⚠️ Warn	Gonzalez & Sahni 1976 verified; Lenstra year slightly off (1977 vs 1978); no concrete complexity bound
Well-written	❌ Fail	Example is a sketch with uniform processing times; decision formulation conflicts with optimization directive (#765); not round-trip testable

Overall: 2 passed, 1 warning, 1 failed

Re-check after previous report from 2026-03-12. Previously: 3 passed, 0 warnings, 1 failed. Now: 2 passed, 1 warning, 1 failed — Correctness downgraded to Warn due to missing concrete complexity bound (not just a well-written issue).

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Usefulness

OpenShopScheduling does not exist in the current codebase (pred show OpenShopScheduling fails). The problem is a classic NP-complete scheduling problem from Garey & Johnson (A5 SS14). Rule issue #481 (Partition -> Open-Shop Scheduling) provides a concrete connection to the existing reduction graph via Partition, which is already implemented.

Non-trivial

This is genuinely distinct from existing scheduling models:

• vs FlowShopScheduling: In flow-shop, all jobs visit machines in the same fixed order. In open-shop, each job's machine order is a decision variable.
• vs JobShopScheduling: In job-shop, each job has a fixed (but job-specific) machine routing. In open-shop, there is no prescribed ordering at all.
• vs MultiprocessorScheduling: In multiprocessor scheduling, each job has a single task on one machine. Open-shop has one task per machine per job.

The feasibility constraints (job-level non-overlap across machines + machine-level non-overlap) with free task ordering are unique.

Correctness

References verified:

• Gonzalez & Sahni, 1976: "Open Shop Scheduling to Minimize Finish Time," JACM 23(4):665-679. NP-completeness for m >= 3 by reduction from Partition confirmed. Polynomial algorithm for m = 2 confirmed. Polynomial preemptive algorithm confirmed.
• Lenstra strong NP-completeness: The issue cites "[Lenstra, 1977]" but the strong NP-completeness result for arbitrary m is typically attributed to either Lenstra's unpublished 1978 manuscript or Brucker, Lenstra, Rinnooy Kan (1977) "Complexity of machine scheduling problems." The result itself is correct; the year attribution is slightly off. Minor inaccuracy — not a failure.
• Garey & Johnson, 1979: A5 SS14 classification confirmed.

Missing concrete complexity bound: The Complexity section says "exact approaches use branch-and-bound, ILP, or constraint programming" but provides no specific worst-case bound usable in declare_variants!. For comparison, JobShopScheduling uses factorial(num_tasks) and FlowShopScheduling uses factorial(num_jobs). A brute-force bound for open-shop would be something like factorial(num_jobs)^num_machines (permutations of jobs on each machine). This needs to be specified.

Well-written

Failures:

1. Example is not fully worked. The schedule is described as a "sketch" — "Jobs {J1, J5} fill gaps on one side" and "Jobs {J2, J3, J4} fill gaps on the other side" without explicit start times. I constructed and verified a concrete valid schedule (see below), but the issue must provide this level of detail:

   Job	M1	M2	M3
   J1	[10,14)	[20,24)	[0,4)
   J2	[20,25)	[0,5)	[10,15)
   J3	[25,28)	[5,8)	[15,18)
   J4	[28,30)	[8,10)	[18,20)
   J5	[14,20)	[24,30)	[4,10)
   J6	[0,10)	[10,20)	[20,30)

2. Uniform processing times. Every job has identical processing times across all 3 machines (J1: 4/4/4, J2: 5/5/5, etc.). This collapses the open-shop structure into a special case where machine assignment order does not matter much. The example should use varied processing times (e.g., J1: 4 on M1, 2 on M2, 5 on M3) to illustrate why open-shop ordering freedom matters.

3. Not round-trip testable as optimization. Total processing time per machine equals the deadline (30 = 4+5+3+2+6+10). This means ANY feasible schedule achieves makespan exactly 30 — there is only one possible objective value. A round-trip test for Value = Min<usize> needs multiple feasible schedules with different makespans so that only a correct reduction finds the true optimum.

4. Decision vs optimization mismatch. The issue body defines a decision problem (with deadline D and YES/NO answer) but the implementation note (Upgrade 17 decision models to optimization versions #765) requires building as optimization (Value = Min<usize>, minimize makespan). The schema still includes a deadline field that should be removed. The definition, variables, schema, and example all need rewriting for the optimization formulation.

5. Variable domain vague for dims(). The domain is stated as {0, 1, ..., D - max_length} for start-time variables, but this is the decision formulation. For the optimization model, a permutation-based encoding (like JobShopScheduling's Lehmer-code approach) would be more appropriate and needs specification.

Recommendations

• Rewrite the entire issue for the optimization formulation: remove deadline from schema, change objective to "minimize makespan," update variables to use permutation encoding.
• Use non-uniform processing times in the example to showcase open-shop flexibility.
• Provide a concrete complexity string for declare_variants! (e.g., factorial(num_jobs)^num_machines as brute-force bound).
• Fully work out the example with explicit start times and verify all constraints.
• Design the example so that the optimal makespan is strictly less than the trivial upper bound, with at least 2 suboptimal feasible makespans.

[isPANN]

Issue Quality Check — Model (Re-check)

Check	Result	Details
Usefulness	✅ Pass	Novel problem not yet in reduction graph; rule #481 (Partition -> OpenShopScheduling) planned
Non-trivial	✅ Pass	Distinct from JobShopScheduling, FlowShopScheduling, and MultiprocessorScheduling
Correctness	⚠️ Warn	Gonzalez & Sahni 1976 verified; Lenstra year slightly off; no concrete complexity bound for declare_variants!
Well-written	❌ Fail	Example sketch with uniform processing times; decision formulation conflicts with optimization directive (#765); not round-trip testable

Overall: 2 passed, 1 warning, 1 failed

Re-check #3 (previous checks: 2026-03-12 by @zazabap, 2026-03-29 by @isPANN). No changes to the issue body since last check. Previously: 2P/1W/1F -> Now: 2P/1W/1F (unchanged). All findings from the 2026-03-29 re-check remain valid.

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Usefulness

OpenShopScheduling does not exist in the codebase (pred show OpenShopScheduling returns "Unknown problem"). It is a classic NP-complete scheduling problem (Garey & Johnson A5 SS14). Rule issue #481 (Partition -> Open-Shop Scheduling) provides a concrete connection to the reduction graph via Partition. The problem is also distinct from the existing FlowShopScheduling and JobShopScheduling models.

Non-trivial

This is genuinely distinct from all existing scheduling models:

• vs FlowShopScheduling: Flow-shop has a fixed machine order for all jobs; open-shop has free task ordering.
• vs JobShopScheduling: Job-shop has job-specific fixed machine routings; open-shop has no prescribed ordering.
• vs MultiprocessorScheduling: Each job has a single task; open-shop has one task per machine per job.

The feasibility constraints (job-level non-overlap across machines + machine-level non-overlap, with free task ordering) are unique.

Correctness

References verified:

• Gonzalez & Sahni, 1976: "Open Shop Scheduling to Minimize Finish Time," JACM 23(4):665-679. Confirmed: NP-completeness for m >= 3 by reduction from Partition; polynomial for m = 2; polynomial preemptive algorithm.
• Lenstra strong NP-completeness: The issue cites "[Lenstra, 1977]". The result is correct, but the attribution is imprecise -- the strong NP-completeness for arbitrary m is typically from Lenstra's unpublished manuscript (sometimes cited as 1977, sometimes 1978) or from Brucker, Lenstra & Rinnooy Kan (1977) "Complexity of Machine Scheduling Problems," Annals of Discrete Mathematics 1:343-362, which is already in the project bibliography. Minor inaccuracy, not a failure.
• Garey & Johnson, 1979: A5 SS14 classification confirmed.

Missing concrete complexity bound: The Complexity section discusses NP-completeness but provides no specific worst-case bound usable in declare_variants!. Existing scheduling models use bounds like factorial(num_tasks) (JobShopScheduling) or factorial(num_jobs) (FlowShopScheduling). A brute-force bound for open-shop could be factorial(num_jobs)^num_machines (permutations of jobs on each machine). This must be specified.

Well-written

Failures (unchanged from previous check):

1. Example is not fully worked. The schedule is described as a "sketch" with vague statements like "Jobs {J_1, J_5} fill gaps on one side." A valid example must show explicit start times for every job on every machine and verify all three constraint types. (Python verification confirmed the schedule from the previous check's worked example IS valid with makespan 30, but the issue body itself still lacks this detail.)

2. Uniform processing times do not exercise open-shop structure. Every job has identical processing times across all 3 machines (J1: 4/4/4, J2: 5/5/5, etc.). Python verification confirms this collapses the open-shop structure -- since each job takes the same time on every machine, the machine assignment order is irrelevant. The example should use varied processing times to illustrate why open-shop ordering freedom matters.

3. Not round-trip testable for optimization. Python verification confirms: total processing time per machine = 30 = D, so ANY feasible schedule achieves makespan exactly 30. There is only one possible objective value, making the instance useless for testing Value = Min<usize> where we need multiple feasible makespans to validate correctness.

4. Decision vs optimization mismatch. The issue body defines a decision problem (deadline D, YES/NO answer) but the implementation note (Upgrade 17 decision models to optimization versions #765) requires Value = Min<usize> (minimize makespan). The schema still includes a deadline field that should be removed. The definition, variables, schema, and example all need rewriting for the optimization formulation.

5. Variable domain vague for dims(). The domain is stated as {0, 1, ..., D - max_length} for start-time variables in the decision formulation. For the optimization model, a permutation-based encoding (like JobShopScheduling) would be more appropriate.

Recommendations

• Rewrite the entire issue for the optimization formulation: remove deadline from schema, change objective to "minimize makespan," update variables to use permutation encoding.
• Use non-uniform processing times in the example (e.g., J1: 4 on M1, 2 on M2, 5 on M3) to showcase open-shop flexibility.
• Provide a concrete complexity string for declare_variants! (e.g., factorial(num_jobs)^num_machines).
• Fully work out the example with explicit start times and constraint verification.
• Design the example so the optimal makespan is strictly less than the trivial upper bound, with at least 2 suboptimal feasible makespans.

[isPANN]

Fix-issue changelog

• Replaced sketch example (uniform processing times, single makespan) with non-uniform 4-job, 3-machine instance
• New example: optimal makespan 11 (4/13824 orderings), 13 distinct makespan values, verified by brute-force
• Reformulated as optimization (removed deadline field, Value = Min<usize>)
• Changed processing_times type from Vec<Vec<u64>> to Vec<Vec<usize>> for consistency
• Added getter methods (num_machines(), num_jobs())
• Added complexity string factorial(num_jobs)^num_machines for declare_variants!
• Added ILP solvability and direct ILP crossref
• Added explicit start times and job-overlap verification in example
• Removed "Unverified" markers from verified content

Applied by /fix-issue.