Why it matters

Vertex cover approximates classic NP-hard. Understanding shapes approximation theory.

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The architecture

Pick any edge.

Add both endpoints.

Remove incident.

Repeat.

Vertex cover 2-approxPick edgeany uncoveredAdd bothendpointsRemove incidentthen repeatLP-relaxation also gives 2-approx via rounding
Vertex cover.
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How it works end to end

Simple to implement.

Both endpoints ensures coverage.

Matching lower bound: |M| <= OPT.

Cover size <= 2|M| <= 2 OPT.