Sketching Valuation Functions

Abstract

Motivated by the problem of querying and communicating bidders' valuations in combinatorial auctions, we study how well different classes of set functions can be sketched. More formally, let f be a function mapping subsets of some ground set [n] to the non-negative real numbers. We say that f is an α-sketch of f if for every set S, the value f (S) lies between f (S)/α and f (S), and f can be specified by poly(n) bits.We show that for every subadditive function f there exists an α-sketch where α = n 1/2 · O(polylog(n)). Furthermore, we provide an algorithm that finds these sketches with a polynomial number of demand queries. This is essentially the best we can hope for since:1. We show that there exist subadditive functions (in fact, XOS functions) that do not admit an o(n 1/2 ) sketch. (Balcan and Harvey [3] previously showed that there exist functions belonging to the class of substitutes valuations that do not admit an O(n 1/3 ) sketch.) 2. We prove that every deterministic algorithm that accesses the function via value queries only cannot guarantee a sketching ratio better than n 1− .We also show that coverage functions, an interesting subclass of submodular functions, admit arbitrarily good sketches. Finally, we show an interesting connection between sketching and learning. We show that for every class of valuations, if the class admits an α-sketch, then it can be α-approximately learned in the PMAC model of Balcan and Harvey. The bounds we prove are only information-theoretic and do not imply the existence of computationally efficient learning algorithms in general.

Economic efficiency requires interaction

“…On one hand several recent papers show that valuations cannot be "compressed", even approximately, and that any polynomial-length description of subadditive valuations (or even the more restricted XOS valuations) must lose a factor of Θ( √ m) in precision [3,2]. Similar, but somewhat weaker, non-approximation results are also known for the far more restricted subclass of "gross-substitutes" valuations [4] for which exact optimization is possible with polynomial communication.…”

“…2 Noninteractive systems are modeled as simultaneous communication protocols, where all agents simultaneously send messages to a central planner who must decide on the allocation based on these messages alone. Interactive systems may use multiple rounds of communication and we measure the amount of interactiveness of a system by the number of communication rounds.…”

Combinatorial Auctions Do Need Modest Interaction

“…The main difference between this setting and ours is that we are interested in arbitrary distributions of inputs for the bidders which are not necessarily product distributions; as already shown by the strong impossibility results of [9], the aforementioned type of protocols cannot provably exist in our model when input distributions are correalted. Finally, we point out that "incompressability" results are also known for subadditive valuations: any polynomial-length encoding of subadditive valuations must lose Ω( √ m) in precision [4,5]. We refer the interested reader to [9] for a comprehensive summary of related work and further discussion on the role of interaction in markets.…”

Gross substitutability: An algorithmic survey

“…The approximability between other classes of valuations has been extensively studied [9,17,14,3], yet very little is known about gross substitutes.…”

“…2 ), where T is the cost of evaluating the valuation function on any given set; and finding the minimum mean weight cycle, which takes time O((n + m) 3 ). This gives an overall running time of…”