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Suppose we have a randomized algorithm that takes a graph G and color k as inputs and provides yes if the graph is k-colorable and no with probability $p(n)$ if it's not k-colorable, where $n$ is the number of vertices. What is the consequence if someone comes with a proof that $\displaystyle \lim_{n \rightarrow \infty} p(n) = 1$.

Does it imply that graph coloring in BPP or ZPP?

N.B. I am familiar with randomized algorithms, NP, NP-completeness, and polynomial reductions as they were covered in Algorithms, Automata, and Languages. However, I didn't go too deep into complexity theory. In my understanding answer to the question is "if its true then graph coloring is in BPP."

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The complexity class RP consists of problems which have a randomized algorithm, running in polynomial time, with the following properties:

  • If the answer is No, the algorithm always answers No.
  • If the answer is Yes, the algorithm answers Yes with probability at least $1/2$ (or at least $1/\mathit{poly}(n)$, or at least $1-1/\exp(\mathit{poly}(n))$ — the resulting class is the same).

You have a similar promise with Yes and No reversed, so the complexity class is coRP.

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  • $\begingroup$ So can we take it further? graph coloring is in NP-complete. So does it not imply that $NP \subseteq co-RP$. $\endgroup$ Commented Apr 17, 2022 at 20:49
  • $\begingroup$ @Yuvai A language L is in BPP if and only if there exists a probabilistic Turing machine M, such that 1. M runs for polynomial time on all inputs 2. For all x in L, M outputs 1 with probability greater than or equal to 2/3 [which is in this case 1] 3 . For all x not in L, M outputs 1 with probability less than or equal to 1/3. [here for all x probability is not < 1/3 but it is $\sim_{n} 1$. $\endgroup$ Commented Apr 17, 2022 at 20:59
  • $\begingroup$ So my basic question is, for finite instances does the probability of false-negative matters or the asymptotic probability of false-negative we have to consider? $\endgroup$ Commented Apr 17, 2022 at 21:05
  • $\begingroup$ Since the probability can be boosted as I indicate in my answer, it doesn’t matter as long as it is in the range indicated in my answer. $\endgroup$ Commented Apr 18, 2022 at 3:33
  • $\begingroup$ OKK. So @Yuvai if graph coloring in coRP that implies $NP \subseteq coRP$. Am I getting correct? $\endgroup$ Commented Apr 18, 2022 at 11:34

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