Why it matters

PTAS / FPTAS classify approximation power. Understanding shapes algorithm goals.

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The architecture

PTAS: for each epsilon > 0, poly-time algorithm.

FPTAS: polynomial in 1/epsilon too.

Approximation schemesPTASpoly(n) per epsilonFPTASpoly(n, 1/epsilon)Both give(1+eps)-approxKnapsack has FPTAS; Euclidean TSP has PTAS; both are ideal outcomes
Schemes.
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How it works end to end

Knapsack: FPTAS by rounding values.

Euclidean TSP: PTAS by Arora.

APX-hard: no PTAS.