Combinatorial Auctions Do Need Modest Interaction

Abstract

We study the necessity of interaction for obtaining efficient allocations in combinatorial auctions with subadditive bidders. This problem was originally introduced by Dobzinski, Nisan, and Oren [9] as the following simple market scenario: m items are to be allocated among n bidders in a distributed setting where bidders valuations are private and hence communication is needed to obtain an efficient allocation. The communication happens in rounds: in each round, each bidder, simultaneously with others, broadcasts a message to all parties involved. At the end, the central planner computes an allocation solely based on the communicated messages. Dobzinski et al. [9] showed that (at least some) interaction is necessary for obtaining any efficient allocation: no non-interactive (1-round) protocol with polynomial communication (in the number of items and bidders) can achieve approximation ratio better than Ω(m 1/4 ), while for any r ≥ 1, there exists r-round protocols that achieve O(r · m 1/r+1 ) approximation with polynomial communication; in particular, O(log m) rounds of interaction suffice to obtain an (almost) efficient allocation, i.e., a polylog(m)-approximation.A natural question at this point is to identify the "right" level of interaction (i.e., number of rounds) necessary to obtain an efficient allocation. In this paper, we resolve this question by providing an almost tight round-approximation tradeoff for this problem: we show that for any r ≥ 1, any r-round protocol that uses poly(m, n) bits of communication can only approximate the social welfare up to a factor of Ω( 1 r · m 1/2r+1 ). This in particular implies that Ω( log m log log m ) rounds of interaction are necessary for obtaining any efficient allocation (i.e., a constant or even a polylog(m)-approximation) in these markets. Our work builds on the recent multi-party round-elimination technique of Alon, Nisan, Raz, and Weinstein [2] -used to prove similar-inspirit lower bounds for round-approximation tradeoff in unit-demand (matching) markets -and settles an open question posed by Dobzinski et al. [9] and Alon et al. [2].

Improved Truthful Mechanisms for Combinatorial Auctions with Submodular Bidders

“…Thus, in an optimal allocation of M to N mech , no item have price more than ψ max and also the total contribution of items with price smaller than ψ min is negligible, hence we can safely ignore them. This in turn implies that Assumption 1 holds and by Theorem 1, PriceLearningMechanism(N mech , M ) outputs an allocation with welfare ALG mech such that E [ALG mech ] ≥ OPT mech · Ω 1 (log log m) 3 . Moreover, by the choice of N mech , we have E [OPT mech ] = OPT/2.…”

“…Main Result. There exists a universally truthful mechanism for combinatorial auctions with submodular valuations that achieves an approximation ratio of O((log log m) 3 ) to the social welfare in expectation using polynomial number of value and demand queries.…”

Polynomial pass lower bounds for graph streaming algorithms

“…Summing up the LHS and RHS in Eq (8) and Eq (9), finalizes the proof. Lemma (Restatement of Lemma 4.7).…”

“…Similar-in-spirit round/communication tradeoffs for distributed computation of many graph and related problems have also been studied in the literature [7,8,11,32,34,51]. For example, [34] proves an Ω( log n log log n ) round lower bound for protocols with low communication that can approximate matchings in a communication model in which players correspond to vertices of an n-vertex graph.…”

Tight Bounds on the Round Complexity of the Distributed Maximum Coverage Problem

“…In establishing Result 1, we introduce a general framework for proving communication complexity lower bounds for bounded round protocols in the distributed coordinator model. This framework, formally introduced in Section 4, captures many of the existing multi-party communication complexity lower bounds in the literature for bounded-round protocols including [12,13,34,51] (for one round a.k.a simultaneous protocols), and [7,8] (for multi-round protocols). We believe our framework will prove useful for establishing distributed lower bound results for other problems, and is thus interesting in its own right.…”