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Oracles are used in complexity theory for lower and upper bounds in the decision tree model. Using oracles you can tell if the proof for certain problems (like P vs NP) relativizes or not. If it relat …

It is believed that $BPP$ has no complete problems. Even for $BPP^O$ for a suitable oracle $O$ it is believed not to have complete problems, unless P=BPP. I wonder if the class MA (the randomized vers …

Normally a 2-sided error randomized algorithm will have some constant error $\varepsilon < 1/2$. We know that we can replace the error term for any inverse polynomial. And the inverse polynomial can b …

A different nice example is a variant of the subset sum problem called Pigeonhole Subset Sum. Given $n$ positive integers with sum less than $2^n-1$, find two disjoint nonempty subsets whose sums are …

Let $\mathbb{F}_q$ be a finite field with characteristic $p$ and $p < q$ (i.e. not a prime field). Let $D\subseteq \mathbb{F}_q$ be a some set with $|D|=n$. Find a non-empty subset $\{x_1,\dots,x_k\} …