Quality control in sublinear time: a case study via random graphs

Many algorithms are designed to work well on average over inputs. When running such an algorithm on an arbitrary input, we must ask: Can we trust the algorithm on this input? We identify a new class of algorithmic problems addressing this, which we call "Quality Control Problems." These problems are specified by a (positive, real-valued) "quality function" $\rho$ and a distribution $D$ such that, with high probability, a sample drawn from $D$ is "high quality," meaning its $\rho$-value is near $1$. The goal is to accept inputs $x \sim D$ and reject potentially adversarially generated inputs $x$ with $\rho(x)$ far from $1$. The objective of quality control is thus weaker than either component problem: testing for "$\rho(x) \approx 1$" or testing if $x \sim D$, and offers the possibility of more efficient algorithms.