Why it matters

Randomized rounding gives approximation for IP. Understanding shapes rounding techniques.

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The architecture

Solve LP relaxation.

For each variable x_i (fractional): set to 1 with probability x_i.

Expected: matches LP.

Randomized roundingSolve LP relaxationfractional solutionRound randomlyx_i probabilityExpected value≈ LP optimumChernoff bounds guarantee concentration around expected
Rounding technique.
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How it works end to end

Chernoff: probability of deviation exponentially small.

Constraint satisfaction: repeat until feasible.

Applications: max-SAT, set cover, network design.