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He's not. Any program code written today that solves an NP-hard problem without exploiting special properties of the input data will take exponential time. If w

by sqrt17 16y ago

He's not. Any program code written today that solves an NP-hard problem without exploiting special properties of the input data will take exponential time. If we discover a smart way to do NP-hard problems in polynomial time, we'll have a better solution for the problem, but the old program code will stay the same. If you want an analogy: sorting an array is clearly a task that can be solved in O(n log n) time, but if you write a bubble sort implementation (or any other O(n^2) algorithm), the program will still take O(n^2) time.

swolchok 16y ago
Right conclusion, not the most direct argument. NP is a proper subset of EXPTIME; there are problems that must take exponential time to complete. Wikipedia says that asking whether a Turing-machine halts in k steps is EXPTIME-complete (you have to execute k steps and the input is of size log k), but if you're willing to fudge a bit and consider function problems, listing all paths in a graph (or something even simpler, like "list all n-bit binary numbers")is going to take exponential time in the worst case.
SamReidHughes 16y ago
Any program code written today that solves an NP-hard problem without exploiting special properties of the input data will take exponential time. If we discover a smart way to do NP-hard problems in polynomial time, we'll have a better solution for the problem, but the old program code will stay the same. You could write a program that runs in polynomial time, but be unable to prove that it does. Then, later, you could find a proof that the program runs in polynomial time. In fact, if P=NP, you can write such a program right now: for each string s in the enumeration of all strings { if (s = p + q where (p is a program and q is a proof that p solves 3SAT in polynomial time)) { run p on the input; return; } } Also, you could interleave the execution of this program with one that solves 3SAT in exponential time. Then you'll have a program that works in any case, and that solves 3SAT in polynomial time if P = NP.

by sqrt17 16y ago

swolchok 16y ago

Right conclusion, not the most direct argument. NP is a proper subset of EXPTIME; there are problems that must take exponential time to complete. Wikipedia says that asking whether a Turing-machine halts in k steps is EXPTIME-complete (you have to execute k steps and the input is of size log k), but if you're willing to fudge a bit and consider function problems, listing all paths in a graph (or something even simpler, like "list all n-bit binary numbers")is going to take exponential time in the worst case.

SamReidHughes 16y ago

Any program code written today that solves an NP-hard problem without exploiting special properties of the input data will take exponential time. If we discover a smart way to do NP-hard problems in polynomial time, we'll have a better solution for the problem, but the old program code will stay the same. You could write a program that runs in polynomial time, but be unable to prove that it does. Then, later, you could find a proof that the program runs in polynomial time. In fact, if P=NP, you can write such a program right now: for each string s in the enumeration of all strings { if (s = p + q where (p is a program and q is a proof that p solves 3SAT in polynomial time)) { run p on the input; return; } } Also, you could interleave the execution of this program with one that solves 3SAT in exponential time. Then you'll have a program that works in any case, and that solves 3SAT in polynomial time if P = NP.