Let $G$ be an undirected, bounded degree graph
with $n$ vertices. Fix a finite graph $H$, and suppose one must remove $\varepsilon n$ edges from $G$ to make it $H$-minor free (for some small constant $\varepsilon > 0$). We give an $n^{1/2+o(1)}$-time randomized procedure that, with high probability, finds an ... more >>>

A central question in derandomization is whether randomized logspace (RL) equals deterministic logspace (L). To show that RL=L, it suffices to construct explicit pseudorandom generators (PRGs) that fool polynomial-size read-once (oblivious) branching programs (roBPs). Starting with the work of Nisan, pseudorandom generators with seed-length $O(\log^2 n)$ were constructed. Unfortunately, ... more >>>

We study bounded depth $(\min, +)$ formulas computing the shortest path polynomial. For depth $2d$ with $d \geq 2$, we obtain lower bounds parametrized by certain fan-in restrictions on all $+$ gates except those at the bottom level. For depth $4$, in two regimes of the parameter, the bounds are ... more >>>