Why it matters

Parameterized complexity provides exact algorithms for restricted cases. Understanding shapes design.

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The architecture

Parameter k: e.g., solution size, treewidth.

FPT: f(k) * poly(n) time.

Kernelization: reduce to size f(k).

Parameterized complexityChoose parameterk = solution sizeFPT algorithmf(k) * poly(n)Kernelizationreduce to size f(k)Vertex cover FPT with O(1.28^k) time; better than 2^n brute force
Parameterized.
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How it works end to end

Vertex cover: O(1.28^k) FPT.

Treewidth-based: many NP-hard problems FPT.

W-hierarchy: harder classes.