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Notwithstanding Friedgut's theorem about $k$-SAT, while we lack techniques to get to negligible $\epsilon$ for small $k$, it seems more useful to talk about the satisfiability threshold ($\alpha - \ep …

A classic problem in probability theory is to express the probability of an event in terms of more specific events. In the simplest case, one can say $P[A \cup B] = P[A] + P[B] - P[A \cap B]$. Let's …

Edit: I am leaving this answer as is (for now) to illustrate the messy process of proving theorems, something that is left out of published papers. The core intuition here is that it is enough to foc …

The second paragraph of RJK's response deserves more detail. Let $\phi$ be a formula in conjunctive normal form, with m clauses, n variables, and at most k variables per clause. Suppose we want to de …

I am still unclear about the precise objective you want to optimise over, but you could look at Peter Brucker, Andreas Drexl, Rolf Möhring, Klaus Neumann, and Erwin Pesch, Resource-constrained proje …