A strongly polynomial algorithm for linear exchange markets

Garg, Jugal; Végh, László A.

doi:10.1145/3313276.3316340

Cited by 18 publications

(12 citation statements)

References 55 publications

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“…In computer science, much work has been devoted to the question of computing exact or approximate equilibria of different markets, which are special cases of the general Arrow-Debreu market that we study. There are several works that developed polynomial-time algorithms for finding or approximating equilibria for some classes of utility functions, e.g., see [Devanur et al, 2008;Jain et al, 2003;Jain, 2007;Duan and Mehlhorn, 2015;Duan et al, 2016;Garg and Végh, 2019;Garg et al, 2004Garg et al, , 2015. For more complex utility functions, besides the results that we mentioned earlier, the approximate equilibrium computation problem for additively separable piecewise linear concave (SPLC) functions was shown to be PPAD-complete by [Vazirani and Yannakakis, 2011;Chen et al, 2009a], where the approximation notion is a "weak approximation" in the market clearing and utility-maximization conditions, see [Scarf, 1967].…”

Section: Related Workmentioning

confidence: 99%

FIXP-membership via Convex Optimization: Games, Cakes, and Markets

Filos-Ratsikas¹,

Hansen²,

Høgh³

et al. 2021

Preprint

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We introduce a new technique for proving membership of problems in FIXP -the class capturing the complexity of computing a fixed-point of an algebraic circuit. Our technique constructs a "pseudogate" which can be used as a black box when building FIXP circuits. This pseudogate, which we term the "OPT-gate", can solve most convex optimization problems. Using the OPT-gate, we prove new FIXP-membership results, and we generalize and simplify several known results from the literature on fair division, game theory and competitive markets.In particular, we prove complexity results for two classic problems: computing a market equilibrium in the Arrow-Debreu model with general concave utilities is in FIXP, and computing an envy-free division of a cake with general valuations is FIXP-complete. We further showcase the wide applicability of our technique, by using it to obtain simplified proofs and extensions of known FIXP-membership results for equilibrium computation for various types of strategic games, as well as the pseudomarket mechanism of Hylland and Zeckhauser.

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Section: Related Workmentioning

confidence: 99%

FIXP-membership via Convex Optimization: Games, Cakes, and Markets

Filos-Ratsikas¹,

Hansen²,

Høgh³

et al. 2021

Preprint

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“…Recently, strongly polynomial algorithms were discovered for the Markov decision problem with a xed discount rate [Ye05,Ye11], minimum-cost ow problems with separable convex objectives [Vég12], generalized ow maximization [Vég17, OV20] and computing market equilibriums for linear exchange markets [GV19]. We refer to [GV19] for more references on strongly polynomial algorithms for related market problems.…”

Section: Further Related Workmentioning

confidence: 99%

Minimizing Convex Functions with Rational Minimizers

Jiang¹

2020

Preprint

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Given a separation oracle SO for a convex function f that has an integral minimizer inside a box with radius R, we show how to e ciently nd a minimizer of f using at most O(n(n+log(R))) calls to SO. When the set of minimizers of f has integral extreme points, our algorithm outputs an integral minimizer of f . This improves upon the previously best oracle complexity of O(n 2 (n + log(R))) obtained by an elegant application of simultaneous diophantine approximation due to Schrijver, Prog. Comb. Opt. 1984, Springer 1988] over thirty years ago. We conjecture that our oracle complexity is tight up to constant factors.Our result immediately implies a strongly polynomial algorithm for the Submodular Function Minimization problem that makes at most O(n 3 ) calls to an evaluation oracle. This improves upon the previously best O(n 3 log 2 (n)) oracle complexity for strongly polynomial algorithms given in [Lee, Sidford and Wong, FOCS 2015] and [Dadush, Végh and Zambelli, SODA 2018], and an exponential time algorithm with oracle complexity O(n 3 log(n)) given in the former work, answering two open problems posted therein.Our result is achieved by an application of the LLL algorithm [Lenstra, Lenstra and Lovász, Math. Ann. 1982] for the shortest lattice vector problem. We show how an approximately shortest vector of certain lattice can be used to reduce the dimension of the problem, and how the oracle complexity of such a procedure is advantageous compared with the Grötschel-Lovász-Schrijver approach that uses simultaneous diophantine approximation. Our analysis of the oracle complexity is based on a potential function that captures simultaneously the size of the search set and the density of the lattice. To achieve the O(n 2 ) term in the oracle complexity, technical ingredients from convex geometry are applied.

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“…Properties of MBB graphs have been crucial in recent algorithmic progress on approximating the Nash Social Welfare, e.g. Cole and Gkatzelis [8], Garg et al [15], Chaudhury et al [7], as well as in computing equilibria in Arrow-Debreu exchange markets (Garg and Végh [14]), but beyond the similarity in the definition we are unaware of any technical overlap.…”

Section: Related Work and Roadmapmentioning

confidence: 99%

Fairness-Efficiency Tradeoffs in Dynamic Fair Division

Zeng

Psomas

2019

Preprint

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We investigate the tradeoffs between fairness and efficiency when allocating indivisible items over time. Suppose T items arrive over time and must be allocated upon arrival, immediately and irrevocably, to one of n agents. Agent i assigns a value v it ∈ [0, 1] to the t-th item to arrive and has an additive valuation function. If the values are chosen by an adaptive adversary, that gets to see the (random) allocations of items 1 through t − 1 before choosing v it , it is known that the algorithm that minimizes maximum pairwise envy simply allocates each item uniformly at random; the maximum pairwise envy is then sublinear in T , namely Õ( T /n). If the values are independently and identically drawn from an adversarially chosen distribution D, it is also known that, under some mild conditions on D, allocating to the agent with the highest value -a Pareto efficient allocation -is envy-free with high probability.In this paper we study fairness-efficiency tradeoffs in this setting and provide matching upper and lower bounds under a spectrum of progressively stronger adversaries. On one hand we show that, even against a non-adaptive adversary, there is no algorithm with sublinear maximum pairwise envy that Pareto dominates the simple algorithm that allocates each item uniformly at random. On the other hand, under a slightly weaker adversary regime where item values are drawn from a known distribution and are independent with respect to time, i.e. v it is independent of v i t but possibly correlated with v ît , optimal (in isolation) efficiency is compatible with optimal (in isolation) fairness. That is, we give an algorithm that is Pareto efficient expost and is simultaneously optimal with respect to fairness: for each pair of agents i and j, either i envies j by at most one item (a prominent fairness notion), or i does not envy j with high probability. En route, we prove a structural (and constructive) result about allocations of divisible items that might be of independent interest: there always exists a Pareto efficient divisible allocation where each agent i either strictly prefers her own bundle to the bundle of agent j, or, if she is indifferent, then i and j have identical allocations and the same value (up to multiplicative factors) for all the items that are allocated to them.

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A strongly polynomial algorithm for linear exchange markets

Cited by 18 publications

References 55 publications

FIXP-membership via Convex Optimization: Games, Cakes, and Markets

FIXP-membership via Convex Optimization: Games, Cakes, and Markets

Minimizing Convex Functions with Rational Minimizers

Fairness-Efficiency Tradeoffs in Dynamic Fair Division

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