role-algorithms:computational-complexity

Computational Complexity

When to use

• Classifying a problem as P, NP-complete, or NP-hard before choosing an algorithm
• Proving a problem is NP-complete via polynomial-time reduction
• Selecting between exact, approximation, FPT, or heuristic approaches
• Designing approximation algorithms with provable guarantees (PTAS, FPTAS)
• Deciding when randomized algorithms (Las Vegas vs Monte Carlo) are appropriate
• Identifying when a parameter makes an otherwise hard problem tractable (FPT)

Core principles

1. Classify before solving — knowing a problem is NP-hard changes the entire strategy
2. Reduction direction matters — reduce FROM a known hard problem TO yours to prove hardness
3. Approximation ratio is a guarantee, not a hope — an unproven heuristic is not an approximation algorithm
4. Small parameter = FPT opportunity — exponential in k is fine when k ≤ 20
5. Amplify randomized correctness — repeat Monte Carlo k times to push error below 2^(-k)

Reference Files

• references/complexity-classes.md — P, NP, NP-complete, NP-hard, PSPACE definitions; common NPC problems; reduction technique step-by-step; recognizing NP-hard problems in practice
• references/approximation-and-fpt.md — approximation ratios, classic results table, PTAS/FPTAS definitions, inapproximability bounds, FPT algorithms, kernelization, treewidth parameter
• references/randomized-and-heuristics.md — Las Vegas vs Monte Carlo, amplification, MCMC, local search, simulated annealing, genetic algorithms, when-to-use decision table