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In graph Frontier
Computational ComplexityComputer ScienceTheory of ComputationEstablished

NP-completeness

An NP-complete problem is among the hardest in NP: a fast solution to one would solve them all, and none is known.

NP-completenessComputational complexity theoryComplexity Class NPPolynomial-Time ReductionComputational ComplexityReduction (complexity)Approximation algorithmPCP theoremComputability theory

At a glance

Type
computation
Disciplines 3
Computational ComplexityComputer ScienceTheory of Computation
Mental models 0
—
Role in the graph
Local hub

Key signals

Cross-disciplinary reach
Disciplines
3
  • Computational Complexity
  • Computer Science
  • Theory of Computation
Evidence & development

Dependencies

What this concept builds on and what it makes possible — derived from the atlas’s dependency, causal and structural relations, not from every related edge.

Foundations · builds on

NP-completenessrequiresPolynomial-Time ReductionEstablished

Open Polynomial-Time Reduction →

NP-Completeness requires Polynomial-Time Reduction.

Mechanism: NP-completeness is defined via polynomial-time reductions from all NP problems.

NP-completenessdepends onReduction (complexity)Established

Open Reduction (complexity) →

NP-Completeness depends on Reduction.

System context

NP-completenessis part ofComplexity Class NPEstablished

Open Complexity Class NP →

NP-Completeness is a part of Complexity Class NP.

NP-completenessis part ofComputational ComplexityEstablished

Open Computational Complexity →

NP-Completeness is a part of Computational Complexity.

NP-completenessis part ofComputational complexity theoryEstablished

Open Computational complexity theory →

NP-completeness is a complexity-theory idea.

Mechanism: It marks the hardest problems in NP: solve one quickly and you solve them all, which no one knows how to do.

Structural role & consequence

Interpreted from the current atlas graph — what the connections mean, not just how many there are.

  • Currently dark in the atlas: no key date stored · 8 of 8 of its relations lack claim-level evidence.

    atlas representation · Describes the current Thinking OS representation, not the state of the world.

  • Builds on 2 foundations (requires / depends-on / derived-from / emerges-from).

    structural · Structural graph analysis — not a claim of importance, causation or history.

  • Structural neighbourhood: 7 → 17 → 40 concepts reachable within 3 hops.

    structural · Structural reach — being reachable is not the same as being understood.

  • All 7 of its relationships stay within its own discipline — a field-specific concept in the current atlas.

    structural · Structural graph analysis — not a claim of importance, causation or history.

0%

cross-field
7 within-field, 0 cross-field

0 of 8 relations carry evidence · concept has a verified source

Strengths & constraints

Constraints

  • Evidence coverage currently thin in the atlas — few of its relationships carry claim-level evidence. atlas representation
  • No dated history stored — the atlas records no key date for this concept. atlas representation

Conditions

  • Its dependency reading rests on 2 foundation relations. structural
  • Read structurally — most of its relationships carry no external evidence yet, so claims here are graph-derived. structural

Dependency radial

What this concept builds on (left) and what it makes possible (right) — derived from dependency and causal relations.

Polynomial-Time Reducti…Reduction (complexity)NP-completeness◀ builds onenables ▶

Seen through each discipline

How this concept sits in each of its fields — derived from its real connections in the graph, not asserted.

Computational Complexity

Through this lens it connects to Complexity Class NP, Polynomial-Time Reduction, Reduction (complexity) and Reduction (complexity).

NP-completeness through the Computational Complexity lens →

Computer Science

Through this lens it connects to Computational complexity theory, Reduction (complexity), Reduction (complexity) and Approximation algorithm.

NP-completeness through the Computer Science lens →

Theory of Computation

Through this lens it connects to Computational Complexity, Reduction (complexity) and Reduction (complexity).

NP-completeness through the Theory of Computation lens →

Concepts that look related but are not yet connected here — candidates for a connection to reason about, not established links.

The scientific picture
  • Connects 7 other ideas across 3 disciplines.
  • A local hub: unusually many ideas converge here.
  • Most of its connections are of the “Kind & structure” kind.

Derived from the graph’s real structure — observations, not a score.

Sources