Why it matters

Matroids explain when greedy works. Understanding shapes algorithmic intuition.

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The architecture

Ground set + independent subsets.

Properties: hereditary + exchange.

Matroid axiomsGround setelementsIndependent setsclosed under subsetsExchangeswap element for biggerKruskal's MST is greedy on graphic matroid; explains why it works
Matroid structure.
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How it works end to end

Examples: uniform matroid, graphic matroid (spanning forest), matric matroid (linearly independent columns).

Weighted matroid intersection: still polynomial.

Applications: MST, scheduling, network design.