patternMinor
Decision problem that can be verified but not run in n^2 time
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Problem
A much much weaker idea similar to the P=NP question, is there a decision problem that can be verified in $O(n^2)$ time, but it can be proven that there is no algorithm that decides it $O(n^2)$ time?
Solution
The answer depends a lot on the model of computation.
On one extreme: in the decision tree model, many lower bounds can be proven. For instance, consider the 3SUM problem. You can verify an alleged solution to the 3SUM problem in $O(n)$ time, but it's conjectured that no algorithm can solve the 3SUM problem in $O(n^{2-\epsilon})$ time for any $\epsilon>0$. One can prove this holds in some version of the algebraic decision tree model; one can prove that any algorithm to solve it in this model must take $\Omega(n^2)$ time. This provides a gap of $O(n^2)$ vs $O(n)$ for solving vs verifying solutions, if the verifier is limited to the algebraic decision tree model. This is a pretty limited result; in this case, the model means that we're restricting attention to algorithms of a particular form. So, it doesn't rule out the possibility of some other algorithm (that does something weird) being faster.
On the other extreme: if we allow arbitrary (non-uniform) boolean circuits, then there is no explicitly stated function $f$ on $n$ bits where we can currently prove that every circuit for computing $f$ needs $\ge 3.1n$ gates. In other words, this is saying that we have no clue how to prove lower bounds for circuits. Roughly speaking, we have no known result of a problem where we can prove that solving it requires circuits of size $\omega(n)$. See, e.g., https://cstheory.stackexchange.com/a/8005/5038 and https://cstheory.stackexchange.com/q/21400/5038. So, given our current state of knowledge, we have no hope of proving a result like that if the model of computation is unrestricted boolean circuits.
On one extreme: in the decision tree model, many lower bounds can be proven. For instance, consider the 3SUM problem. You can verify an alleged solution to the 3SUM problem in $O(n)$ time, but it's conjectured that no algorithm can solve the 3SUM problem in $O(n^{2-\epsilon})$ time for any $\epsilon>0$. One can prove this holds in some version of the algebraic decision tree model; one can prove that any algorithm to solve it in this model must take $\Omega(n^2)$ time. This provides a gap of $O(n^2)$ vs $O(n)$ for solving vs verifying solutions, if the verifier is limited to the algebraic decision tree model. This is a pretty limited result; in this case, the model means that we're restricting attention to algorithms of a particular form. So, it doesn't rule out the possibility of some other algorithm (that does something weird) being faster.
On the other extreme: if we allow arbitrary (non-uniform) boolean circuits, then there is no explicitly stated function $f$ on $n$ bits where we can currently prove that every circuit for computing $f$ needs $\ge 3.1n$ gates. In other words, this is saying that we have no clue how to prove lower bounds for circuits. Roughly speaking, we have no known result of a problem where we can prove that solving it requires circuits of size $\omega(n)$. See, e.g., https://cstheory.stackexchange.com/a/8005/5038 and https://cstheory.stackexchange.com/q/21400/5038. So, given our current state of knowledge, we have no hope of proving a result like that if the model of computation is unrestricted boolean circuits.
Context
StackExchange Computer Science Q#104894, answer score: 5
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