What is the formal definition of an approximation algorithm?
Conditions
- Applies to NP-hard optimization problems
- Runs in polynomial time
Reasoning, step by step
- Identify the problem as an NP-hard optimization problem.
- Design an algorithm that runs in polynomial time.
- Ensure the algorithm finds a near-optimal solution.
- Provide a mathematical guarantee on the solution quality.
Example
The video states: "So formally speaking, approximation algorithms are polynomial time algorithms used to find near-optimal solutions for NP-hard optimization problems."
Common misconceptions
- Thinking approximation algorithms are just random guesses without guarantees.
- Assuming they can always find the exact optimal solution in polynomial time.
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Conditions: Applies to the design of approximation algorithms for NP-hard problems.
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Conditions: Problem is an NP-hard optimization problem; Seeking an exact solution
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Conditions: Comparing standard approximation classes for optimization problems.
An approximation bound is a guaranteed multiplicative limit on how much worse the algorithm's result can be than the true optimum in the worst case.
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