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

## TL;DR

This paper develops a theory for minimal linear sketches that approximate valuation functions over ^n with small error, analyzing their dimensions for various function types and connecting these results to streaming and distributed algorithms.

## Contribution

It introduces a general framework for linear sketching of valuation functions over ^n, providing tight bounds on sketch dimensions for multiple function classes and extending to streaming and distributed models.

## Key findings

- Characterized sketch dimensions for additive, submodular, and matroid functions.
- Established tight bounds for most function classes analyzed.
- Extended results to streaming and distributed computation contexts.

## Abstract

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, $\alpha$-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.