Why it matters

PTAS/FPTAS give best approximation guarantees. Understanding shapes what's provably achievable.

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The architecture

PTAS: (1+ε) approx, poly(n) time.

FPTAS: poly(n, 1/ε).

Approximation classesPTAS(1+ε), poly(n)FPTASpoly(n, 1/ε)APX-hardno PTASAPX-hard problems: no PTAS unless P=NP (e.g., max-cut)
PTAS hierarchy.
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How it works end to end

FPTAS for knapsack: rounding based.

PTAS for Euclidean TSP.

APX-hardness: rule out PTAS.

Practical trade-off: smaller ε = slower.