Why it matters

NP-hard problems need approximate solutions. Understanding shapes practical NP-hard approach.

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The architecture

Approximation ratio: alg cost / optimal cost.

Guarantee: worst-case bound.

Approximation ratiosApproximation ratioα ≥ 1Guaranteeworst-casePTAS / FPTASarbitrarily close to optPTAS: (1+ε) approximation for any ε > 0
Approximation classes.
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How it works end to end

Classic: 2-approx TSP metric via MST.

Set cover: ln n greedy.

Vertex cover: 2-approx.

Hardness of approximation: some problems can't be approximated well (PCP theorem).