Approximate $\mathbb{F}_2$-Sketching of Valuation Functions

We study the problem of constructing a linear sketch of minimum dimension that allows approximation of a given real-valued function $f \colon \mathbb{F}_2^n \rightarrow \mathbb R$ with small expected squared error. We develop a general theory of linear sketching for such functions through which we analyze their dimension for most commonly studied types of valuation functions: additive, budget-additive, coverage, $α$-Lipschitz submodular and matroid rank functions. This gives a characterization of how many bits of information have to be stored about the input $x$ so that one can compute $f$ under additive updates to its coordinates. Our results are tight in most cases and we also give extensions to the distributional version of the problem where the input $x \in \mathbb{F}_2^n$ is generated uniformly at random. Using known connections with dynamic streaming algorithms, both upper and lower bounds on dimension obtained in our work extend to the space complexity of algorithms evaluating $f(x)$ under long sequences of additive updates to the input $x$ presented as a stream. Similar results hold for simultaneous communication in a distributed setting.