Strong NP-completeness¶

In computational complexity, strong NP-completeness is a property of computational problems that is a special case of NP-completeness.

Version: v1 · 2026-09-28 · History
Domain-specific #: 12324
Domain group: Applied Sciences & Engineering
Origin domain: Computer Science & Software Engineering
Subdomain: Computational Complexity → Computer Science & Software Engineering

Core Idea¶

Strong NP-completeness is treated here as the recurring computer_science_and_information identity summarized by this source-grounded definition: In computational complexity, strong NP-completeness is a property of computational problems that is a special case of NP-completeness.

In computational complexity, strong NP-completeness is a property of computational problems that is a special case of NP-completeness. A general computational problem may have numerical parameters. For example, the input to the bin packing problem is a list of objects of specific sizes and a size for the bins that must contain the objects—these object sizes and bin size are numerical parameters.

A problem is said to be strongly NP-complete (NP-complete in the strong sense), if it remains NP-complete even when all of its numerical parameters are bounded by a polynomial in the length of the input. A problem is said to be strongly NP-hard if a strongly NP-complete problem has a pseudo-polynomial reduction to it. This pseudo-polynomial reduction is more restrictive than the usual poly-time reduction used for NP-hardness proofs.

For Strong NP-completeness, the abstraction is narrower than the article's general subject matter: a positive case must preserve In computational complexity, strong NP-completeness is a property of computational problems that is a special case of NP-completeness. Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in computer_science_and_information, which is why this identity is domain-specific rather than prime.

Structural Signature¶

Sig role-phrases:

Defining carrier — A problem is said to be strongly NP-complete (NP-complete in the strong sense), if it remains NP-complete even when all of its numerical parameters are bounded by a polynomial in the length of the input.
Constitutive relation — In particular, the pseudo-polynomial reduction cannot output a numerical parameter that is not polynomially bounded by the size and value of numbers in the input.
Operating condition — If we redefine the problem to have the parameters given in unary notation, then the parameters must be bounded by the input size.
Recognition evidence — Thus the version of bin packing where the object and bin sizes are integers bounded by a polynomial remains NP-complete, while the corresponding version of the Knapsack problem can be solved in pseudo-polynomial time by dynamic programming.
Admissible variation — In computational complexity, strong NP-completeness is a property of computational problems that is a special case of NP-completeness.
Characteristic consequence — For example, the input to the bin packing problem is a list of objects of specific sizes and a size for the bins that must contain the objects—these object sizes and bin size are numerical parameters.
Failure boundary — A problem is said to be strongly NP-hard if a strongly NP-complete problem has a pseudo-polynomial reduction to it.

What It Is Not¶

Not the whole field of computer_science_and_information. The node requires the specific identity stated by In computational complexity, strong NP-completeness is a property of computational problems that is a special case of NP-completeness.
Not an over-broad reading. However, the converse fails: e.g. if P does not equal NP, knapsack with two constraints is not strongly NP-hard, but has no FPTAS even when the optimal objective is polynomially bounded.
Not an over-broad reading. In particular, the pseudo-polynomial reduction cannot output a numerical parameter that is not polynomially bounded by the size and value of numbers in the input.
Not an over-broad reading. In computational complexity, strong NP-completeness is a property of computational problems that is a special case of NP-completeness.
Not automatically Complete (complexity). Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

Strong NP-completeness applies literally inside computer_science_and_information wherever the source-defined carrier and relation can be established. Its documented habitats include:

Documented setting. This pseudo-polynomial reduction is more restrictive than the usual poly-time reduction used for NP-hardness proofs.
Documented setting. From a theoretical perspective any strongly NP-hard optimization problem with a polynomially bounded objective function cannot have a fully polynomial-time approximation scheme (or FPTAS) unless P = NP.
Documented setting. Some strongly NP-complete problems may still be easy to solve on average, but it's more likely that difficult instances will be encountered in practice.
Documented setting. In computational complexity, strong NP-completeness is a property of computational problems that is a special case of NP-completeness.
Documented setting. For example, the input to the bin packing problem is a list of objects of specific sizes and a size for the bins that must contain the objects—these object sizes and bin size are numerical parameters.
Documented setting. A problem is said to be strongly NP-complete (NP-complete in the strong sense), if it remains NP-complete even when all of its numerical parameters are bounded by a polynomial in the length of the input.

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

Clarity¶

A clear use of Strong NP-completeness names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In computational complexity, strong NP-completeness is a property of computational problems that is a special case of NP-completeness. The strongest recognition evidence in the frozen account is: Thus the version of bin packing where the object and bin sizes are integers bounded by a polynomial remains NP-complete, while the corresponding version of the Knapsack problem can be solved in pseudo-polynomial time by dynamic programming. A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification However, the converse fails: e.g. if P does not equal NP, knapsack with two constraints is not strongly NP-hard, but has no FPTAS even when the optimal objective is polynomially bounded. so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity¶

Strong NP-completeness compresses multiple computer_science_and_information details into a stable diagnostic relation. The source shows both the central mechanism—in particular, the pseudo-polynomial reduction cannot output a numerical parameter that is not polynomially bounded by the size and value of numbers in the input.—and the practical consequence—for example, the input to the bin packing problem is a list of objects of specific sizes and a size for the bins that must contain the objects—these object sizes and bin size are numerical parameters. 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 computer_science_and_information entities to which the claim applies.
State the relation. Use the source-grounded identity: In computational complexity, strong NP-completeness is a property of computational problems that is a special case of NP-completeness.
Check operation and conditions. If we redefine the problem to have the parameters given in unary notation, then the parameters must be bounded by the input size.
Demand recognition evidence. Thus the version of bin packing where the object and bin sizes are integers bounded by a polynomial remains NP-complete, while the corresponding version of the Knapsack problem can be solved in pseudo-polynomial time by dynamic programming.
Test variation. Change an implementation or setting while preserving in computational complexity, strong NP-completeness is a property of computational problems that is a special case of NP-completeness.
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 Classification.

Knowledge Transfer¶

Within the home domain. Knowledge about Strong NP-completeness transfers literally when a new case preserves the same carrier type, relation, and recognition test. This pseudo-polynomial reduction is more restrictive than the usual poly-time reduction used for NP-hardness proofs. From a theoretical perspective any strongly NP-hard optimization problem with a polynomially bounded objective function cannot have a fully polynomial-time approximation scheme (or FPTAS) unless P = NP.

Beyond the home domain. No canonical parent is asserted for Strong NP-completeness. 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¶

In computational complexity, strong NP-completeness is a property of computational problems that is a special case of NP-completeness. 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 computational complexity, strong NP-completeness is a property of computational problems that is a special case of NP-completeness; recognition evidence → Thus the version of bin packing where the object and bin sizes are integers bounded by a polynomial remains NP-complete, while the corresponding version of the Knapsack problem can be solved in pseudo-polynomial time by dynamic programming

Applied / In Practice¶

For example, the input to the bin packing problem is a list of objects of specific sizes and a size for the bins that must contain the objects—these object sizes and bin size are numerical parameters. 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 → the applied context; invariant → In computational complexity, strong NP-completeness is a property of computational problems that is a special case of NP-completeness; boundary → the case exits the class when however, the converse fails: e.g. if P does not equal NP, knapsack with two constraints is not strongly NP-hard, but has no FPTAS even when the optimal objective is polynomially bounded

Structural Tensions¶

T1 — Stable identity versus admissible variation. However, the converse fails: e.g. if P does not equal NP, knapsack with two constraints is not strongly NP-hard, but has no FPTAS even when the optimal objective is polynomially bounded. 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. In particular, the pseudo-polynomial reduction cannot output a numerical parameter that is not polynomially bounded by the size and value of numbers in the input. 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. In computational complexity, strong NP-completeness is a property of computational problems that is a special case of NP-completeness. 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. For example, the input to the bin packing problem is a list of objects of specific sizes and a size for the bins that must contain the objects—these object sizes and bin size are numerical parameters. 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. A problem is said to be strongly NP-complete (NP-complete in the strong sense), if it remains NP-complete even when all of its numerical parameters are bounded by a polynomial in the length of the input. 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 Strong NP-completeness literally, co-instantiate Classification, or only resemble it?

T6 — Autonomy versus reduction. In particular, the pseudo-polynomial reduction cannot output a numerical parameter that is not polynomially bounded by the size and value of numbers in the input. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Strong NP-completeness distinguish that the broader parent Classification leaves together?

Structural–Framed Character¶

Strong NP-completeness is structural-leaning. Its structural side is the repeatable organization summarized by In computational complexity, strong NP-completeness is a property of computational problems that is a special case of NP-completeness. Its framed side is the computer_science_and_information 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: If we redefine the problem to have the parameters given in unary notation, then the parameters must be bounded by the input size. Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Classification. 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 computational complexity, strong NP-completeness is a property of computational problems that is a special case of NP-completeness. 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: A problem is said to be strongly NP-complete (NP-complete in the strong sense), if it remains NP-complete even when all of its numerical parameters are bounded by a polynomial in the length of the input. In particular, the pseudo-polynomial reduction cannot output a numerical parameter that is not polynomially bounded by the size and value of numbers in the input. It further constrains recognition and variation through: If we redefine the problem to have the parameters given in unary notation, then the parameters must be bounded by the input size. Thus the version of bin packing where the object and bin sizes are integers bounded by a polynomial remains NP-complete, while the corresponding version of the Knapsack problem can be solved in pseudo-polynomial time by dynamic programming.

What is domain-bound. computer science and information supplies the operative entities, technical vocabulary, warrants, and exceptions that make Strong NP-completeness literal. Its documented scope includes the condition that This pseudo-polynomial reduction is more restrictive than the usual poly-time reduction used for NP-hardness proofs. Another bounded application condition is that From a theoretical perspective any strongly NP-hard optimization problem with a polynomially bounded objective function cannot have a fully polynomial-time approximation scheme (or FPTAS) unless P = NP. 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—In computational complexity, strong NP-completeness is a property of computational problems that is a special case of NP-completeness.—and future graph densification may discover a defensible relation only if it preserves that boundary.

This entry presupposes Complexity Class.

Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for Strong NP-completeness. The reviewed identity is: In computational complexity, strong NP-completeness is a property of computational problems that is a special case of NP-completeness. 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 Strong NP-completeness Domain-specific

Parents (1) — more general patterns this builds on

Strong NP-completeness presupposes Complexity Class Domain-specific

Strong NP-completeness is defined through NP-completeness under restrictions on numerical encoding and therefore presupposes the class structure.

Neighborhood in Abstraction Space¶

Strong NP-completeness sits in a moderately populated region (58^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Unclustered & Miscellaneous (2551 abstractions)

Nearest neighbors

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

Not to Be Confused With¶

Classification. The parent omits the specialist differentia. Tell: Can the case establish In computational complexity, strong NP-completeness is a property of computational problems that is a special case of NP-completeness?
Complete (complexity). Complete (complexity) denotes notion of the "hardest" or "most general" problem in a complexity class in computing and information systems. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Strongly-polynomial time. A complexity notion requiring polynomially many arithmetic operations independent of numeric magnitudes. Tell: Which entry's carrier, operation, and failure condition are satisfied?
FPT (complexity class). FPT (complexity class) names a recurring computing and information systems identity with specialized roles and obligations not carried by the frozen neighbors. 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 Strong NP-completeness remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside computer_science_and_information lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Classification?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Strong_NP-completeness (revision 1361867404).
Preserved source candidate: https://cstheory.stackexchange.com/questions/7382/can-strong-np-hardness-really-be-shown-using-plain-polytime-reductions
Preserved source candidate: https://archive.org/details/computersintract0000gare/page/

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.