Why it matters

Clique reduction is illustrative NP-hard proof. Understanding shapes technique.

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The architecture

Vertices per literal in clauses.

Edges between compatible literals in different clauses.

Clique of size m iff 3-SAT satisfiable.

3-SAT -> Clique reductionVerticesone per literalEdgescompatible cross-clauseClique size miff 3-SAT satKarp 1972; illustrates gadget-based NP-hardness proofs
Clique reduction.
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How it works end to end

Clause i has 3 literal vertices.

Edge between (i, x) and (j, y) if i != j and not contradictory.

m clauses, need m-clique.