The graph homomorphism problem HOM is: given an $n$-vertex source graph $G$ and an $h$-vertex target graph $H$, is there a mapping from $V(G)$ to $V(H)$ that preserves edges? A straightforward brute-force algorithm for HOM has running time $O(2^{n \log h})$ and it is known that, under ETH, there are ... more >>>

The Orthogonal Vectors Problem (OV$_{n,d}$) takes as input two sets $A,B$ each containing $n$ $d$-dimensional Boolean vectors, and outputs $1$ if and only if there exists $a \in A$ and $b \in B$ such that $a$ and $b$ are orthogonal. The OV conjecture states that for every $\varepsilon > 0$, ... more >>>

For every $n,s \geq 1$, we construct a matrix tuple $(A_1,\ldots,A_n) \in \mathrm{M}_s(\mathbb{Z})^n$ in deterministic $\mathrm{poly}(n,s)$ time such that every noncommutative polynomial $$f \in \mathbb{C}\langle x_1,x_2,\ldots,x_n\rangle$$ of sparsity at most $s$ satisfies $f = 0$ if and only if $f(A_1,A_2,\ldots,A_n) = 0$. The bit complexity of the entries in ... more >>>