We study the parallel complexity of computing the arboricity of a graph, defined as the minimum number of forests into which its edges can be partitioned.
For graphs of bounded treewidth, we present a simple dynamic programming–based parallel algorithm that constructs an optimal partition of the edges into forests.
For ... more >>>

A depth-4 algebraic circuit with top fan-in $k$ and bottom fan-in $2$ is a circuit $\Phi$ of the form $\Phi = \sum_{i=1}^k \prod_{j=1}^{m_i} Q_{ij}$, where the polynomials $Q_{ij} \in \mathbb{K}[x_1, \ldots, x_n]$ have degree at most $2$.
The class of all such circuits is denoted by $\Sigma^k \Pi \Sigma ... more >>>

For a Boolean function $f:\{0,1\}^n\to\{0,1\}$, the higher-order Boolean derivative $D_S f$ computes the parity of $f$ over each $S$-dimensional subcube. We prove that $D_S f\equiv 1$ exactly when $S$ is a maximal monomial support in the algebraic normal form of $f$. This correspondence motivates the derivative certificate depth $\Delta_\partial(f)$, defined ... more >>>