Faster and Simpler Sketches of Valuation Functions
Keren Cohavi, Shahar Dobzinski

TL;DR
This paper introduces simpler, faster algorithms for sketching valuation functions like submodular and subadditive functions, improving efficiency while maintaining optimal approximation bounds.
Contribution
It provides a non-geometric, simpler proof for the existence of good sketches and develops faster algorithms matching known approximation bounds.
Findings
Faster algorithms for submodular function sketching with 7(n^{3/2}) queries.
Efficient sketching of subadditive functions with O(n) demand and value queries.
Simplified proof technique avoids complex geometric constructions.
Abstract
We present fast algorithms for sketching valuation functions. Let () be some ground set and be a function. We say that is an -sketch of if for every set we have that and can be described in bits. Goemans et al. [SODA'09] showed that if is submodular then there exists an -sketch that can be constructed using polynomially many value queries (this is the best possible, as Balcan and Harvey [STOC'11] show that no submodular function admit an -sketch). Based on their work, Balcan et al. [COLT'12] and Badanidiyuru et al. [SODA'12] show that if is subadditive then there exists an -sketch that can be constructed using polynomially many demand queries. All…
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