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Sure, in some cases (the easy ones) we can solve NP-complete problems. However, tons of useful instances of NP-complete problems are not particularly close to

Sure, in some cases (the easy ones) we can solve NP-complete problems. However, tons of useful instances of NP-complete problems are not particularly close to being solved. We can even characterize fairly well when the instances become hard (for instance, for graph problems the treewidth seems relevant). I see absolutely no evidence that we're anywhere close to constructing a polynomial-time algorithm for any NP-complete problem. If anything, we are slowly finally making some progress on proving P != NP because we are finally getting some nontrivial superlinear bounds on these problems (thus far, the biggest obstacle to proving P != NP has been the difficulty of proving anything about lower bounds on runtime that isn't directly tied to space consumption). Seriously, if you're interested in this approach ("well, in practice we can solve the problem if [insert thing here] is bounded"), check out the field of parameterized complexity. It's very popular in the database area. It also reveals some very sharp limits to tractability for these problems in practice, not just in theory.

lotsoflumens 10y ago
>> (thus far, the biggest obstacle to proving P != NP has been the difficulty of proving anything about lower bounds on runtime that isn't directly tied to space consumption) I probably don't understand the proofs of these bounds - it seems to me though, that any lower bound without some relationship to the time or space to input the problem into some kind of computer, implies a belief that an instance might be solvable in-place, in some physical system. I have some doubts about that ....
- Jweb_Guru 10y ago
  To be clearer, a trivial lower bound is something like "the input is of size n and for any n-1 length yes-instance you can find n-length extensions that are no-instances, so you have to check all n elements." It doesn't tell us anything interesting about the worst-case problem complexity. What's been surprising is how hard it is to make meaningful lower-bound statements stronger than that (well, not surprising to people who work on this stuff for a living, I guess). But that doesn't make it more likely that P = NP, really, except in the sense that we (obviously) haven't characterized the entire space of problems in P yet. To more specifically address your "in-place" concerns, you can often formulate the problem in terms of circuit complexity--circuits are inherently parallel, so minimum circuit depth is a good proxy for a required number of steps, and it's also generally agreed that if you can design a circuit to do something it's physically realizable (even though that's arguably not always true, especially in cases with very wide fanout). So a (somewhat) promising direction has been to try to find circuit complexity lower bounds for problems, especially nontrivial ones.

lotsoflumens 10y ago

>> (thus far, the biggest obstacle to proving P != NP has been the difficulty of proving anything about lower bounds on runtime that isn't directly tied to space consumption) I probably don't understand the proofs of these bounds - it seems to me though, that any lower bound without some relationship to the time or space to input the problem into some kind of computer, implies a belief that an instance might be solvable in-place, in some physical system. I have some doubts about that ....

Jweb_Guru 10y ago
To be clearer, a trivial lower bound is something like "the input is of size n and for any n-1 length yes-instance you can find n-length extensions that are no-instances, so you have to check all n elements." It doesn't tell us anything interesting about the worst-case problem complexity. What's been surprising is how hard it is to make meaningful lower-bound statements stronger than that (well, not surprising to people who work on this stuff for a living, I guess). But that doesn't make it more likely that P = NP, really, except in the sense that we (obviously) haven't characterized the entire space of problems in P yet. To more specifically address your "in-place" concerns, you can often formulate the problem in terms of circuit complexity--circuits are inherently parallel, so minimum circuit depth is a good proxy for a required number of steps, and it's also generally agreed that if you can design a circuit to do something it's physically realizable (even though that's arguably not always true, especially in cases with very wide fanout). So a (somewhat) promising direction has been to try to find circuit complexity lower bounds for problems, especially nontrivial ones.