# Approximate 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 F_2^n → 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 ∈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.

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