# Does randomness buy us anything inside P?

Let $\mathsf{BPTIME}(f(n))$ be the class of the decision problems having a bounded two-sided error randomized algorithm running in time $O(f(n))$.

Do we know of any problem $Q \in \mathsf{P}$ such that $Q \in \mathsf{BPTIME}(n^k)$ but $Q \not \in \mathsf{DTIME}(n^k)$? Is its non-existence proven?

This question was asked on cs.SE here, but did not get a satisfactory answer.

• (1) BPP(f(n)) is usually denoted as BPTIME(f(n)). (2) In the computational complexity setting, I believe that this is open. (Many examples are known in the query complexity and the communication complexity settings.) (3) If its nonexistence were already proved, then we would already know that P=BPP. – Tsuyoshi Ito Jun 17 '12 at 12:30
• By the way, in the question on cs.stackexchange.com, you have some misunderstanding about the relation between BPTIME and ZPTIME, and that might be part of the reason you have not received a satisfactory answer. – Tsuyoshi Ito Jun 17 '12 at 12:38
• @TsuyoshiIto Thanks, I don't agree that if we prove inexistence then we know $P=BPP$, I am restricting the setting to problems in $P$. Maybe, $BPTIME(n^k) \cap P = DTIME(n^k)$, while $BPTIME(n^k) \neq DTIME(n^k)$ in general, am I missing something? Could you also please point out my misunderstanding about $BPTIME$ and $ZPTIME$, maybe I missed a satisfactory answer indeed.. – aelguindy Jun 17 '12 at 13:01
• Your question does not say that you restrict problem Q to be inside P. If that is your intent, please edit the question. – Tsuyoshi Ito Jun 17 '12 at 13:27
• For approximating the 1-median of a finite metric space with few queries to the distance function, a random point gives a 2-approximation in expectation and a (2+eps)-approx with good probability. But no deterministic algorithm that queries the distance function $o(n^2)$ times can do better than a 4-approximation. [ Chang 2013] – Neal Young Aug 12 '14 at 2:58

Another example is estimating the volume of a polyhedron in high dimensions. There's an unconditional lower bound on deterministic strategies to approximate the volume to even an exponential factor, but there's an FPRAS for the problem.

Update: the relevant paper is (link to PDF):

I. Barany and Z. Furedi. Computing the volume is difficult, Discrete and Computational Geometry 2 (1987), 319-326.

• Could you provide reference for the unconditional lower bound? – T.... Mar 7 '14 at 22:03
• added reference. – Suresh Venkat Mar 8 '14 at 3:29

Problem: An array $A[1..2n]$ consists of $n$ 1s and $n$ 0s. Find an $i$ such that $A[i]$ is 1.

You are allowed to query 'Which number is present in $A[i]$'? Each query takes constant time.

Solution: Randomized Algorithm: Pick a random index $i$ and check if $A[i]$ is 1. Expected number of queries is 2, but any deterministic algorithm must make at least $n$ queries. Therefore, the randomized upper bound is strictly better than the deterministic lower bound in this model.

This is an example from query complexity which Tsuyoshi was referring to in the comment.

• Any deterministic algorithm must make at least $n$ queries in the worst case. – argentpepper Jun 18 '12 at 22:18
• What do you mean by "currently we don't know any non-trivial lower bound proof for any problem in NP (let alone P)" ? – Kristoffer Arnsfelt Hansen Jun 19 '12 at 9:12
• Perhaps I used the word 'non-trivial' sloppily. I meant 'Currently, we can't prove an unconditional lower bound of $\Omega(n^k)$ for $k>0$ for SAT or any problem in NP'. Is that correct? – Jagadish Jun 19 '12 at 17:07
• Well, maybe not for "nice" problems such as SAT; but remember we do have such lower bounds for other problems from the time hierarchy theorems. And the question is not about "nice" problems, but about complexity classes. – Kristoffer Arnsfelt Hansen Jun 19 '12 at 21:44
• Ah, right. I assumed that OP was interested in natural problems. I've edited my answer. – Jagadish Jun 20 '12 at 6:15

Given an $n\times n$ payoff matrix for a zero-sum matrix with payoffs in [0,1], estimate the value of the game within an additive $\epsilon$.

This problem has a randomized algorithm that runs in time $O(n\log^2(n)/\epsilon^2)$, which (provably) no deterministic algorithm can match [GK95].