Why it matters

3-SAT is workhorse of NP-hardness. Understanding shapes reduction technique.

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The architecture

Reduce SAT to 3-SAT via clause splitting.

Reduce 3-SAT to target problem to prove NP-hard.

3-SAT reduction pipelineSATany clause size3-SATexactly 3 literalsReduce to targetprove NP-hardKarp's 3-SAT reduction adds auxiliary vars to split larger clauses; polynomial
3-SAT reduction.
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How it works end to end

SAT to 3-SAT: split clauses adding aux vars.

3-SAT to Clique / VC / TSP standard NP-hard proofs.