Unsolved Problems

Conjecture If $X_1, \dotsc, X_n \geq 0$ are independent random variables with $\mathbb{E}[X_i] \leq \mu$, then $$\mathrm{Pr} \left( \sum X_i - \mathbb...

Problem Is P = NP?...

Conjecture The famous 0-1 Knapsack problem is: Given $a_{1},a_{2},\dots,a_{n}$ and $b$ integers, determine whether or not there are $0-1$ values $x_{...

Problem Given two codes $R,C$, their Tensor Product $R \otimes C$ is the code that consists of the matrices whose rows are codewords of $R$ and whose ...

Problem Let $a_1,a_2,\ldots,a_n$ be natural numbers with $\sum_{i=1}^n a_i < 2^n - 1$. It follows from the pigeon-hole principle that there exist dist...

Question What is the complexity of the following problem? Given $a_1,\dots,a_n; k$, determine whether or not $\sum_i \sqrt{a_i} \leq k.$...

Problem Can $O(n)$-size circuits compute the function $f$ on $\{0,1,2\}^*$ defined inductively by $f(\lambda) = \lambda$, $f(0x) = 0f(x)$, $f(1x) = 1f...

If $p$ is prime and $g,h \in {\mathbb Z}_p^*$, we write $\log_g(h) = n$ if $n \in {\mathbb Z}$ satisfies $g^n = h$. The problem of finding such an int...

Problem Is there a problem that can be computed by a Turing machine in polynomial space and unbounded time but not in polynomial time? More formally, ...

Conjecture One-way functions exist....

Problem Prove unconditionally that $\mathcal{AM}$ $\subseteq$ $\Sigma_2$....

Conjecture Can all problems that can be computed by a probabilistic Turing machine (with error probability < 1/3) in polynomial time be solved by a de...

Conjecture For every rational $\epsilon > 0$ and every rational $\Delta$, there is no polynomial-time algorithm for the following problem. Given is a...