approximation algorithms guarantee: solution within factor α of optimal

Can we always find the best solution quickly?

Imagine you're planning a road trip and trying to find the shortest route. You have several cities to visit, and you want to minimize driving time.

Sometimes, finding the perfect route is too complex, so we use shortcuts that get us close to the ideal path without taking forever.

Example

Instead of calculating every possible route, you use a shortcut that estimates the best route within 10% of the shortest possible time.

Remember this

Approximation algorithms help us find nearly optimal solutions faster when exact solutions are too time-consuming.

Text adapted from Wikipedia, licensed under CC BY-SA 4.0.

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Travelling salesman problem

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Convex optimization

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UMAP is faster than t-SNE

UMAP is faster due to approximate nearest neighbors and cross-entropy optimization

Overlapping subproblems

Ever calculated a huge Fibonacci sequence by hand?

second-order methods (Newton's) converge faster but are expensive: O(n³) per step

Second-order methods converge faster due to quadratic convergence but are expensive due to O(n³) per iteration

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