Uniform-machines scheduling¶

In the specific variant called uniform machine scheduling, some machines are uniformly faster than others.

Version: v1 · 2026-09-28 · History
Domain-specific #: 12702
Domain group: Formal Sciences
Origin domain: Operations Research
Subdomain: Scheduling Theory → Operations Research

Core Idea¶

Uniform-machines scheduling is treated here as the recurring scheduling theory identity summarized by this source-grounded definition: In the specific variant called uniform machine scheduling, some machines are uniformly faster than others.

Uniform machine scheduling (also called uniformly-related machine scheduling or related machine scheduling) is an optimization problem in computer science and operations research. It is a variant of optimal job scheduling. We are given n jobs J 1 , J 2 , ..., J n of varying processing times, which need to be scheduled on m different machines.

The goal is to minimize the makespan - the total time required to execute the schedule. The time that machine i needs in order to process job j is denoted by p i,j . In the general case, the times p i,j are unrelated, and any matrix of positive processing times is possible.

For Uniform-machines scheduling, the abstraction is narrower than the article's general subject matter: a positive case must preserve In the specific variant called uniform machine scheduling, some machines are uniformly faster than others. Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in scheduling theory, which is why this identity is domain-specific rather than prime.

Structural Signature¶

Sig role-phrases:

Defining carrier — The SPT algorithm (Shortest Processing Time First), sorts the jobs by their length, shortest first, and then assigns them to the processor with the earliest end time so far.
Constitutive relation — A constant-factor approximation is attained by the Longest-processing-time-first algorithm (LPT).
Operating condition — The time that machine i needs in order to process job j is denoted by p i,j .
Recognition evidence — Minimizing the weighted average completion time is NP-hard even on identical machines, by reduction from the knapsack problem.
Admissible variation — It is NP-hard even if the number of machines is fixed and at least 2, by reduction from the partition problem.
Characteristic consequence — Minimizing the maximum completion time is NP-hard even for identical machines, by reduction from the partition problem.
Failure boundary — It means that, if a machine reports a higher speed, and all other inputs remain the same, then the total processing time allocated to the machine weakly increases.

What It Is Not¶

Not the whole field of scheduling theory. The node requires the specific identity stated by In the specific variant called uniform machine scheduling, some machines are uniformly faster than others.
Not an over-broad reading. They claim that their algorithms can be easily extended for any number of uniform machines, but do not analyze the run-time in this case.
Not an over-broad reading. They do not present an algorithm for weighted-average completion time on unrelated machines.
Not an over-broad reading. Ambrosio and Auletta proved that the Longest Processing Time algorithm is monotone whenever the machine speeds are powers of some c ≥ 2, but not when c ≤ 1.78.
Not automatically Scheduling. Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

Uniform-machines scheduling applies literally inside scheduling theory wherever the source-defined carrier and relation can be established. Its documented habitats include:

Minimizing the weighted-average completion time. Epstein and Sgall generalized the PTAS for uniform machines to handle more general objective functions.
Minimizing the weighted-average completion time. Instead of minimizing the objective function max(C i ), one can minimize the objective function max(f(C i )), where f is any fixed function.
Minimizing the weighted-average completion time. Similarly, one can minimize the objective function sum(f(C i )).
AlgorithmsMinimizing the average completion time. The SPT algorithm (Shortest Processing Time First), sorts the jobs by their length, shortest first, and then assigns them to the processor with the earliest end time so far.
AlgorithmsMinimizing the average completion time. It runs in time O(n log n), and minimizes the average completion time on identical machines, P|| \sum C_i .
AlgorithmsMinimizing the average completion time. Horowitz and Sahni present an exact algorithm, with run time O(n log m n), for minimizing the average completion time on uniform machines, Q|| \sum C_i .

Outside scheduling theory, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Pattern or should be marked as analogy.

Clarity¶

A clear use of Uniform-machines scheduling names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In the specific variant called uniform machine scheduling, some machines are uniformly faster than others. The strongest recognition evidence in the frozen account is: Minimizing the weighted average completion time is NP-hard even on identical machines, by reduction from the knapsack problem. A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification They claim that their algorithms can be easily extended for any number of uniform machines, but do not analyze the run-time in this case. so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity¶

Uniform-machines scheduling compresses multiple scheduling theory details into a stable diagnostic relation. The source shows both the central mechanism—a constant-factor approximation is attained by the Longest-processing-time-first algorithm (LPT).—and the practical consequence—minimizing the maximum completion time is NP-hard even for identical machines, by reduction from the partition problem. This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.

Abstract Reasoning¶

Type the carrier. Identify the scheduling theory entities to which the claim applies.
State the relation. Use the source-grounded identity: In the specific variant called uniform machine scheduling, some machines are uniformly faster than others.
Check operation and conditions. The time that machine i needs in order to process job j is denoted by p i,j .
Demand recognition evidence. Minimizing the weighted average completion time is NP-hard even on identical machines, by reduction from the knapsack problem.
Test variation. Change an implementation or setting while preserving it is NP-hard even if the number of machines is fixed and at least 2, by reduction from the partition problem.
Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Pattern.

Knowledge Transfer¶

Within the home domain. Knowledge about Uniform-machines scheduling transfers literally when a new case preserves the same carrier type, relation, and recognition test. Epstein and Sgall generalized the PTAS for uniform machines to handle more general objective functions. Instead of minimizing the objective function max(C i ), one can minimize the objective function max(f(C i )), where f is any fixed function.

Beyond the home domain. No canonical parent is asserted for Uniform-machines scheduling. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Examples¶

Canonical¶

For example, take the case of reading user credentials from console, then use it to authenticate, then if authentication is successful display some data on the console. This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.

Mapped back: carrier → the entities in the documented case; operation → In the specific variant called uniform machine scheduling, some machines are uniformly faster than others; recognition evidence → Minimizing the weighted average completion time is NP-hard even on identical machines, by reduction from the knapsack problem

Applied / In Practice¶

They claim that their algorithms can be easily extended for any number of uniform machines, but do not analyze the run-time in this case. The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.

Mapped back: changed setting → Minimizing the weighted-average completion time; invariant → In the specific variant called uniform machine scheduling, some machines are uniformly faster than others; boundary → the case exits the class when they claim that their algorithms can be easily extended for any number of uniform machines, but do not analyze the run-time in this case

Structural Tensions¶

T1 — Stable identity versus admissible variation. They claim that their algorithms can be easily extended for any number of uniform machines, but do not analyze the run-time in this case. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Which changes preserve the defining relation, and which replace it?

T2 — Recognition versus proxy. They do not present an algorithm for weighted-average completion time on unrelated machines. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the cited evidence establish the identity or only a correlated sign?

T3 — Definition versus implementation. Ambrosio and Auletta proved that the Longest Processing Time algorithm is monotone whenever the machine speeds are powers of some c ≥ 2, but not when c ≤ 1.78. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Is the observed implementation constitutive, optional, or merely common?

T4 — Scope versus overextension. In contrast, List scheduling is not monotone for c > 2. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Can every claimed application fill the same typed roles without metaphor?

T5 — Transfer versus domain accent. The SPT algorithm (Shortest Processing Time First), sorts the jobs by their length, shortest first, and then assigns them to the processor with the earliest end time so far. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the receiving case instantiate Uniform-machines scheduling literally, co-instantiate Pattern, or only resemble it?

T6 — Autonomy versus reduction. A constant-factor approximation is attained by the Longest-processing-time-first algorithm (LPT). The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Uniform-machines scheduling distinguish that the broader parent Pattern leaves together?

Structural–Framed Character¶

Uniform-machines scheduling is mixed or framed-leaning. Its structural side is the repeatable organization summarized by In the specific variant called uniform machine scheduling, some machines are uniformly faster than others. Its framed side is the scheduling theory vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.

Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: The time that machine i needs in order to process job j is denoted by p i,j . Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Pattern. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.

Structural Core vs. Domain Accent¶

What is skeletal. In the specific variant called uniform machine scheduling, some machines are uniformly faster than others. The stable skeleton is the typed relation expressed in that definition and the entry's recognition and collapse tests. The source identifies these operative conditions: The SPT algorithm (Shortest Processing Time First), sorts the jobs by their length, shortest first, and then assigns them to the processor with the earliest end time so far. A constant-factor approximation is attained by the Longest-processing-time-first algorithm (LPT). It further constrains recognition and variation through: The time that machine i needs in order to process job j is denoted by p i,j . Minimizing the weighted average completion time is NP-hard even on identical machines, by reduction from the knapsack problem.

What is domain-bound. scheduling theory supplies the operative entities, technical vocabulary, warrants, and exceptions that make Uniform-machines scheduling literal. Its documented scope includes the condition that Epstein and Sgall generalized the PTAS for uniform machines to handle more general objective functions. Another bounded application condition is that Instead of minimizing the objective function max(C i ), one can minimize the objective function max(f(C i )), where f is any fixed function. These are not decorative examples; they determine which carrier and evidence can fill the abstraction's roles.

Why no parent is asserted. Removing those specialist details does not currently yield one live catalog node that is a necessary genus for every instance. The entry is therefore approved as unparented rather than attached by topical resemblance. Its collapse evidence remains specific—It is NP-hard even if the number of machines is fixed and at least 2, by reduction from the partition problem.—and future graph densification may discover a defensible relation only if it preserves that boundary.

This entry is a decomposition of Scheduling.

Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for Uniform-machines scheduling. The reviewed identity is: In the specific variant called uniform machine scheduling, some machines are uniformly faster than others. The accelerated suggestion was declined because topical or lexical similarity does not establish hierarchy; the node is admitted without a parent pending later graph densification.
Related reasoning operations. Evidence, representation, comparison, classification, transformation, or evaluation may participate in particular cases, but participation does not make any one of them a necessary parent of every instance.

Relationships to Other Abstractions¶

Current abstraction Uniform-machines scheduling Domain-specific

Parents (1) — more general patterns this builds on

Uniform-machines scheduling is a decomposition of Scheduling Prime

Uniform-machine scheduling retains assignment and timing under resource constraints while specializing machines by speed.

Hierarchy paths (5) — routes to 3 parentless roots

Uniform-machines scheduling → Scheduling → Allocation → Scarcity → Constraint

Show alternative paths (4)

Neighborhood in Abstraction Space¶

Uniform-machines scheduling sits in a sparse region of the domain-specific corpus (68^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (2551 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Pattern. The parent omits the specialist differentia. Tell: Can the case establish In the specific variant called uniform machine scheduling, some machines are uniformly faster than others?
Scheduling. Organizing tasks over time. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Job-shop scheduling. Order machine-specific operations for jobs with fixed technological precedences so shared machines never overlap and a declared schedule objective is optimized. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Single-machine scheduling. The optimization of job order and timing on one capacity-one resource under declared release, precedence and objective rules. Tell: Which entry's carrier, operation, and failure condition are satisfied?
A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would Uniform-machines scheduling remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside scheduling theory lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Pattern?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Uniform-machines_scheduling (revision 1359825428).
Preserved source candidate: https://epubs.siam.org/doi/abs/10.1137/0217033
Preserved source candidate: https://doi.org/10.1007/s00453-003-1077-7
Preserved source candidate: https://link.springer.com/chapter/10.1007/978-3-540-24749-4_53
Preserved source candidate: https://link.springer.com/chapter/10.1007/978-3-540-31833-0_22
Preserved source candidate: https://link.springer.com/chapter/10.1007/978-3-540-31856-9_6
Preserved source candidate: https://link.springer.com/chapter/10.1007/11561071_55
Preserved source candidate: https://www2.informatik.uni-osnabrueck.de/knust/class/dateien/classes/pqr_nop/pqr_nop/

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.