We study the complexity of Boolean constraint satisfaction problems (CSPs) when the assignment must have Hamming weight in some congruence class modulo $M$, for various choices of the modulus $M$. Due to the known classification of tractable Boolean CSPs, this mainly reduces to the study of three cases: 2SAT, HornSAT, ... more >>>

We show that tree-like resolution is not automatable in time $n^{o(\log n)}$ unless ETH is false. This implies that, under ETH, the algorithm given by Beame and Pitassi (FOCS 1996) that automates tree-like resolution in time $n^{O(\log n)}$ is optimal. We also provide a simpler proof of the result of ... more >>>

We show the following hardness results for monotone learning and approximation of monotone circuit size:

1. Under the Randomised Exponential-Time Hypothesis (rETH), it requires time $n^{\Omega(\log n)}$ to PAC-learn monotone formulas with $n$ input bits and size $s(n) = n$ by monotone circuits of size $n^{(\log n)^{1-\epsilon}}$, for every $\epsilon ... more >>>