Demand Queries with Preprocessing

Feige, Uriel; Jozeph, Shlomo

doi:10.1007/978-3-662-43948-7_40

Cited by 7 publications

(6 citation statements)

References 16 publications

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“…Here we give a size-O(n) LP relaxation for approximating independent set within a factor-O( √ n), which follows directly by work of Feige and Jozeph [21]. Note that this is strictly better than the n 1−ε hardness obtained assuming P NP by [26].…”

Section: Upper Boundsmentioning

confidence: 86%

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No Small Linear Program Approximates Vertex Cover Within a Factor 2 − ɛ

et al. 2018

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The vertex cover problem is one of the most important and intensively studied combinatorial optimization problems. Khot and Regev [32,33] proved that the problem is NP-hard to approximate within a factor 2 − ε, assuming the Unique Games Conjecture (UGC). This is tight because the problem has an easy 2-approximation algorithm. Without resorting to the UGC, the best inapproximability result for the problem is due to Dinur and Safra [17,18]: vertex cover is NP-hard to approximate within a factor 1.3606.We prove the following unconditional result about linear programming (LP) relaxations of the problem: every LP relaxation that approximates vertex cover within a factor 2 − ε has superpolynomially many inequalities. As a direct consequence of our methods, we also establish that LP relaxations (as well as SDP relaxations) that approximate the independent set problem within any constant factor have super-polynomial size. ContributionWe consider the general model of LP relaxations as in [13], see also [10]. Given an n-vertex graph G = (V, E), a system of linear inequalities Ax b in R d , where d ∈ N is arbitrary, defines an LP relaxation of vertex cover (on G) if the following conditions hold:Linear objective: For every vertex-costs c ∈ R V + , we have an affine function (degree-1 polynomial) Consistency: For all vertex coversFor every vertex-costs c ∈ R V + , the LP min{ f c (x) | Ax b} provides a guess on the minimum cost of a vertex cover. This guess is always a lower bound on the optimum.We allow arbitrary computations for writing down the LP, and do not bound the size of the coefficients. We only care about the following two parameters and their relationship: the size of the LP relaxation, defined as the number of inequalities in Ax b, and the (graph-specific) integrality gap which is the worst-case ratio over all vertex-costs between the true optimum and the guess provided by the LP, for this particular graph G and LP relaxation.This framework subsumes the polyhedral-pair approach in extended formulations [8]; see also [43]. We refer the interested reader to the surveys [15,28] for an introduction to extended formulations; see also Section 4 for more details.In this paper, we prove the following result about LP relaxations of vertex cover and, as a byproduct, independent set. 1 Theorem 1.1. For infinitely many values of n, there exists an n-vertex graph G such that: (i) Every size-n o(log n/ log log n) LP relaxation of vertex cover on G has integrality gap 2 − o(1); (ii) Every sizen o(log n/ log log n) LP relaxation of independent set on G has integrality gap ω(1).This solves an open problem that was posed both by Singh [51] and Chan, Lee, Raghavendra and Steurer [13]. In fact, Singh conjectured that every compact (that is, polynomial size), symmetric extended formulation for vertex cover has integrality gap at least 2−ε. We prove that his conjecture holds, even if asymmetric extended formulations are allowed. 2 Our result for the independent set problem is even stronger than Theorem 1.1, as we are also able to rul...

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Section: Upper Boundsmentioning

confidence: 86%

“…Feige and Jozeph [21] made the following observation: Proof. When solving the LP, we may assume x v = y j for all j ∈ [k] and all v ∈ I j .…”

Section: Upper Boundsmentioning

confidence: 97%

No Small Linear Program Approximates Vertex Cover Within a Factor 2 − ɛ

et al. 2018

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“…2 e mechanism proposed in [1] achieves a constant approximation to the optimal social welfare if the size of the initial endowment of each agent is bounded by a constant; otherwise the approximation factor is of logarithmic order. 3 is question also applies to standard demand queries [10], which may be computationally hard to answer or may involve To additionally achieve a mechanism that results in a high social welfare, we exclude some items from trade and introduce randomness into the mechanism. e main idea is to suppose that all the items are available to the set of buyers as in a one-sided auction, and to compute the expected marginal contribution of an item to the social welfare [11] under this assumption.…”

Section: Overview Of the Techniquesmentioning

confidence: 99%

Approximately Efficient Two-Sided Combinatorial Auctions

Colini-Baldeschi

Goldberg

Keijzer

et al. 2017

Proceedings of the 2017 ACM Conference on Economics and Computation

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We develop and extend a line of recent work on the design of mechanisms for two-sided markets. e markets we consider consist of buyers and sellers of a number of items, and the aim of a mechanism is to improve the social welfare by arranging purchases and sales of the items. A mechanism is given prior distributions on the agents' valuations of the items, but not the actual valuations; thus the aim is to maximise the expected social welfare over these distributions. As in previous work, we are interested in the worst-case ratio between the social welfare achieved by a truthful mechanism, and the best social welfare possible.Our main result is an incentive compatible and budget balanced constant-factor approximation mechanism in a se ing where buyers have XOS valuations and sellers' valuations are additive.is is the rst such approximation mechanism for a two-sided market se ing where the agents have combinatorial valuation functions. To achieve this result, we introduce a more general kind of demand query that seems to be needed in this situation. In the simpler case that sellers have unit supply (each having just one item to sell), we give a new mechanism whose welfare guarantee improves on a recent one in the literature. We also introduce a more demanding version of the strong budget balance (SBB) criterion, aimed at ruling out certain "unnatural" transactions satis ed by SBB. We show that the stronger version is satis ed by our mechanisms.

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“…It is shown in [14] that even some submodular functions that have a polynomial representation do not have any polynomial size implementation of demand oracles (not even oracles that answer demand queries approximately), unless N P has polynomial size circuits. In fact, there are such functions that do not have any polynomial size implementation of value oracles (unless N P has polynomial size circuits).…”

Section: B Mph As a Useful Representationmentioning

confidence: 99%

A Unifying Hierarchy of Valuations with Complements and Substitutes

Feige¹,

Feldman²,

Immorlica³

et al. 2014

Preprint

Self Cite

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We introduce a new hierarchy over monotone set functions, that we refer to as MPH (Maximum over Positive Hypergraphs). Levels of the hierarchy correspond to the degree of complementarity in a given function. The highest level of the hierarchy, MPH-m (where m is the total number of items) captures all monotone functions. The lowest level, MPH-1, captures all monotone submodular functions, and more generally, the class of functions known as X OS. Every monotone function that has a positive hypergraph representation of rank k (in the sense defined by Abraham, Babaioff, Dughmi and Roughgarden [EC 2012]) is in MPH-k. Every monotone function that has supermodular degree k (in the sense defined by Feige and Izsak [ITCS 2013]) is in MPH-(k + 1). In both cases, the converse direction does not hold, even in an approximate sense. We present additional results that demonstrate the expressiveness power of MPH-k.One can obtain good approximation ratios for some natural optimization problems, provided that functions are required to lie in low levels of the MPH hierarchy. We present two such applications. One shows that the maximum welfare problem can be approximated within a ratio of k + 1 if all players hold valuation functions in MPH-k. The other is an upper bound of 2k on the price of anarchy of simultaneous first price auctions.Being in MPH-k can be shown to involve two requirements -one is monotonicity and the other is a certain requirement that we refer to as P LE (Positive Lower Envelope). Removing the monotonicity requirement, one obtains the PLE hierarchy over all non-negative set functions (whether monotone or not), which can be fertile ground for further research.

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Demand Queries with Preprocessing

Cited by 7 publications

References 16 publications

No Small Linear Program Approximates Vertex Cover Within a Factor 2 − ɛ

No Small Linear Program Approximates Vertex Cover Within a Factor 2 − ɛ

Approximately Efficient Two-Sided Combinatorial Auctions

A Unifying Hierarchy of Valuations with Complements and Substitutes

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