Does P mean "easy"? P versus NP problem

the graph shows time (average of 100 instances in ms using 933 mhz pentium iii) vs.problem size knapsack problems state-of-the-art specialized algorithm. quadratic fit suggests empirical algorithmic complexity instances 50–10,000 variables o((log(n))).

all of above discussion has assumed p means easy , not in p means hard , assumption known cobham s thesis. common , reasonably accurate assumption in complexity theory; however, has caveats.

first, not true in practice. theoretical polynomial algorithm may have extremely large constant factors or exponents rendering impractical. on other hand, if problem shown np-complete, , if p ≠ np, there may still effective approaches tackling problem in practice. there algorithms many np-complete problems, such knapsack problem, traveling salesman problem , boolean satisfiability problem, can solve optimality many real-world instances in reasonable time. empirical average-case complexity (time vs. problem size) of such algorithms can surprisingly low. example simplex algorithm in linear programming, works surprisingly in practice; despite having exponential worst-case time complexity runs on par best known polynomial-time algorithms.

second, there types of computations not conform turing machine model on p , np defined, such quantum computation , randomized algorithms.