arXiv subjects

Vijay V. Vazirani

Publications and source records attributed to Vijay V. Vazirani.

At least 55 records · Page 3Linked to original sources

arXiv2017

Concave Flow on Small Depth Directed Networks

Small depth networks arise in a variety of network related applications, often in the form of maximum flow and maximum weighted matching. Recent works have generalized such methods to include costs arising from concave functions. In this paper we give an algorithm that takes a depth $D$ network and strictly increasing concave weight functions of flows on the edges and computes a $(1 - ε)$-approximation to the maximum weight flow in time $mD ε^{-1}$ times an overhead that is logarithmic in the various numerical parameters related to the magnitudes of gradients and capacities. Our approach is based on extending the scaling algorithm for approximate maximum weighted matchings by [Duan-Pettie JACM`14] to the setting of small depth networks, and then generalizing it to concave functions. In this more restricted setting of linear weights in the range $[w_{\min}, w_{\max}]$, it produces a $(1 - ε)$-approximation in time $O(mD ε^{-1} \log( w_{\max} /w_{\min}))$. The algorithm combines a variety of tools and provides a unified approach towards several problems involving small depth networks.

cs.DS↗

arXiv2017

A New Class of Combinatorial Markets with Covering Constraints: Algorithms and Applications

We introduce a new class of combinatorial markets in which agents have covering constraints over resources required and are interested in delay minimization. Our market model is applicable to several settings including scheduling, cloud computing, and communicating over a network. This model is quite different from the traditional models, to the extent that neither do the classical equilibrium existence results seem to apply to it nor do any of the efficient algorithmic techniques developed to compute equilibria seem to apply directly. We give a proof of existence of equilibrium and a polynomial time algorithm for finding one, drawing heavily on techniques from LP duality and submodular minimization. We observe that in our market model, the set of equilibrium prices could be a connected, non-convex set. To the best of our knowledge, this is the first natural example of the phenomenon where the set of solutions could have such complicated structure, yet there is a combinatorial polynomial time algorithm to find one. Finally, we show that our model inherits many of the fairness properties of traditional equilibrium models.

cs.GT↗

arXiv2017

Nash Social Welfare for Indivisible Items under Separable, Piecewise-Linear Concave Utilities

Recently Cole and Gkatzelis gave the first constant factor approximation algorithm for the problem of allocating indivisible items to agents, under additive valuations, so as to maximize the Nash Social Welfare. We give constant factor algorithms for a substantial generalization of their problem -- to the case of separable, piecewise-linear concave utility functions. We give two such algorithms, the first using market equilibria and the second using the theory of stable polynomials. In AGT, there is a paucity of methods for the design of mechanisms for the allocation of indivisible goods and the result of Cole and Gkatzelis seemed to be taking a major step towards filling this gap. Our result can be seen as another step in this direction.

cs.GT↗

arXiv2016

A Performance-Based Scheme for Pricing Resources in the Cloud

With the rapid growth of the cloud computing marketplace, the issue of pricing resources in the cloud has been the subject of much study in recent years. In this paper, we identify and study a new issue: how to price resources in the cloud so that the customer's risk is minimized, while at the same time ensuring that the provider accrues his fair share. We do this by correlating the revenue stream of the customer to the prices charged by the provider. We show that our mechanism is incentive compatible in that it is in the best interest of the customer to provide his true revenue as a function of the resources rented. We next add another restriction to the price function, i.e., that it be linear. This removes the distortion that creeps in when the customer has to pay more money for less resources. Our algorithms for both the schemes mentioned above are efficient.

cs.GT↗

arXiv2016

Convex Program Duality, Fisher Markets, and Nash Social Welfare

We study Fisher markets and the problem of maximizing the Nash social welfare (NSW), and show several closely related new results. In particular, we obtain: -- A new integer program for the NSW maximization problem whose fractional relaxation has a bounded integrality gap. In contrast, the natural integer program has an unbounded integrality gap. -- An improved, and tight, factor 2 analysis of the algorithm of [7]; in turn showing that the integrality gap of the above relaxation is at most 2. The approximation factor shown by [7] was $2e^{1/e} \approx 2.89$. -- A lower bound of $e^{1/e}\approx 1.44$ on the integrality gap of this relaxation. -- New convex programs for natural generalizations of linear Fisher markets and proofs that these markets admit rational equilibria. These results were obtained by establishing connections between previously known disparate results, and they help uncover their mathematical underpinnings. We show a formal connection between the convex programs of Eisenberg and Gale and that of Shmyrev, namely that their duals are equivalent up to a change of variables. Both programs capture equilibria of linear Fisher markets. By adding suitable constraints to Shmyrev's program, we obtain a convex program that captures equilibria of the spending-restricted market model defined by [7] in the context of the NSW maximization problem. Further, adding certain integral constraints to this program we get the integer program for the NSW mentioned above. The basic tool we use is convex programming duality. In the special case of convex programs with linear constraints (but convex objectives), we show a particularly simple way of obtaining dual programs, putting it almost at par with linear program duality. This simple way of finding duals has been used subsequently for many other applications.

cs.GT↗

arXiv2016

Mutation, Sexual Reproduction and Survival in Dynamic Environments

A new approach to understanding evolution [Val09], namely viewing it through the lens of computation, has already started yielding new insights, e.g., natural selection under sexual reproduction can be interpreted as the Multiplicative Weight Update (MWU) Algorithm in coordination games played among genes [CLPV14]. Using this machinery, we study the role of mutation in changing environments in the presence of sexual reproduction. Following [WVA05], we model changing environments via a Markov chain, with the states representing environments, each with its own fitness matrix. In this setting, we show that in the absence of mutation, the population goes extinct, but in the presence of mutation, the population survives with positive probability. On the way to proving the above theorem, we need to establish some facts about dynamics in games. We provide the first, to our knowledge, polynomial convergence bound for noisy MWU in a coordination game. Finally, we also show that in static environments, sexual evolution with mutation converges, for any level of mutation.

q-bio.PE↗

arXiv2016

New Convex Programs for Fisher's Market Model and its Generalizations

We present the following results pertaining to Fisher's market model: -We give two natural generalizations of Fisher's market model: In model M_1, sellers can declare an upper bound on the money they wish to earn (and take back their unsold good), and in model M_2, buyers can declare an upper bound on the amount to utility they wish to derive (and take back the unused part of their money). -We derive convex programs for the linear case of these two models by generalizing a convex program due to Shmyrev and the Eisenberg-Gale program, respectively. -We generalize the Arrow-Hurwicz theorem to the linear case of these two models, hence deriving alternate convex programs. -For the special class of convex programs having convex objective functions and linear constraints, we derive a simple set of rules for constructing the dual program (as simple as obtaining the dual of an LP). Using these rules we show a formal relationship between the two seemingly different convex programs for linear Fisher markets, due to Eisenberg-Gale and Shmyrev; the duals of these are the same, upto a change of variables.

cs.GT↗

arXiv2015

Allocation of Divisible Goods under Lexicographic Preferences

We present a simple and natural non-pricing mechanism for allocating divisible goods among strategic agents having lexicographic preferences. Our mechanism has favorable properties of incentive compatibility (strategy-proofness), Pareto efficiency, envy-freeness, and time efficiency.

cs.GT↗

arXiv2014

Settling Some Open Problems on 2-Player Symmetric Nash Equilibria

Over the years, researchers have studied the complexity of several decision versions of Nash equilibrium in (symmetric) two-player games (bimatrix games). To the best of our knowledge, the last remaining open problem of this sort is the following; it was stated by Papadimitriou in 2007: find a non-symmetric Nash equilibrium (NE) in a symmetric game. We show that this problem is NP-complete and the problem of counting the number of non-symmetric NE in a symmetric game is #P-complete. In 2005, Kannan and Theobald defined the "rank of a bimatrix game" represented by matrices (A, B) to be rank(A+B) and asked whether a NE can be computed in rank 1 games in polynomial time. Observe that the rank 0 case is precisely the zero sum case, for which a polynomial time algorithm follows from von Neumann's reduction of such games to linear programming. In 2011, Adsul et. al. obtained an algorithm for rank 1 games; however, it does not solve the case of symmetric rank 1 games. We resolve this problem.

cs.GT↗

arXiv2014

Learning Economic Parameters from Revealed Preferences

A recent line of work, starting with Beigman and Vohra (2006) and Zadimoghaddam and Roth (2012), has addressed the problem of {\em learning} a utility function from revealed preference data. The goal here is to make use of past data describing the purchases of a utility maximizing agent when faced with certain prices and budget constraints in order to produce a hypothesis function that can accurately forecast the {\em future} behavior of the agent. In this work we advance this line of work by providing sample complexity guarantees and efficient algorithms for a number of important classes. By drawing a connection to recent advances in multi-class learning, we provide a computationally efficient algorithm with tight sample complexity guarantees ($Θ(d/ε)$ for the case of $d$ goods) for learning linear utility functions under a linear price model. This solves an open question in Zadimoghaddam and Roth (2012). Our technique yields numerous generalizations including the ability to learn other well-studied classes of utility functions, to deal with a misspecified model, and with non-linear prices.

cs.GT↗

arXiv2013

On Computability of Equilibria in Markets with Production

Although production is an integral part of the Arrow-Debreu market model, most of the work in theoretical computer science has so far concentrated on markets without production, i.e., the exchange economy. This paper takes a significant step towards understanding computational aspects of markets with production. We first define the notion of separable, piecewise-linear concave (SPLC) production by analogy with SPLC utility functions. We then obtain a linear complementarity problem (LCP) formulation that captures exactly the set of equilibria for Arrow-Debreu markets with SPLC utilities and SPLC production, and we give a complementary pivot algorithm for finding an equilibrium. This settles a question asked by Eaves in 1975 of extending his complementary pivot algorithm to markets with production. Since this is a path-following algorithm, we obtain a proof of membership of this problem in PPAD, using Todd, 1976. We also obtain an elementary proof of existence of equilibrium (i.e., without using a fixed point theorem), rationality, and oddness of the number of equilibria. We further give a proof of PPAD-hardness for this problem and also for its restriction to markets with linear utilities and SPLC production. Experiments show that our algorithm runs fast on randomly chosen examples, and unlike previous approaches, it does not suffer from issues of numerical instability. Additionally, it is strongly polynomial when the number of goods or the number of agents and firms is constant. This extends the result of Devanur and Kannan (2008) to markets with production. Finally, we show that an LCP-based approach cannot be extended to PLC (non-separable) production, by constructing an example which has only irrational equilibria.

cs.GT↗

arXiv2013

A Simplification of the MV Matching Algorithm and its Proof

For all practical purposes, the Micali-Vazirani general graph maximum matching algorithm is still the most efficient known algorithm for the problem. The purpose of this paper is to provide a complete proof of correctness of the algorithm in the simplest possible terms; graph-theoretic machinery developed for this purpose also helps simplify the algorithm.

cs.DS↗

arXiv2013

Thrifty Algorithms for Multistage Robust Optimization

We consider a class of multi-stage robust covering problems, where additional information is revealed about the problem instance in each stage, but the cost of taking actions increases. The dilemma for the decision-maker is whether to wait for additional information and risk the inflation, or to take early actions to hedge against rising costs. We study the "k-robust" uncertainty model: in each stage i = 0, 1,...,T, the algorithm is shown some subset of size k_i that completely contains the eventual demands to be covered; here k_1 > k_2 >...> k_T which ensures increasing information over time. The goal is to minimize the cost incurred in the worst-case possible sequence of revelations. For the multistage k-robust set cover problem, we give an O(log m + log n)-approximation algorithm, nearly matching the Ω(log n + log m/loglog m) hardness of approximation even for T=2 stages. Moreover, our algorithm has a useful "thrifty" property: it takes actions on just two stages. We show similar thrifty algorithms for multi-stage k-robust Steiner tree, Steiner forest, and minimum-cut. For these problems our approximation guarantees are O(min{T, log n, log L_{max}), where L_{max} is the maximum inflation over all the stages. We conjecture that these problems also admit O(1)-approximate thrifty algorithms.

cs.DS↗

arXiv2011

Modeling Tiered Pricing in the Internet Transit Market

ISPs are increasingly selling "tiered" contracts, which offer Internet connectivity to wholesale customers in bundles, at rates based on the cost of the links that the traffic in the bundle is traversing. Although providers have already begun to implement and deploy tiered pricing contracts, little is known about how such pricing affects ISPs and their customers. While contracts that sell connectivity on finer granularities improve market efficiency, they are also more costly for ISPs to implement and more difficult for customers to understand. In this work we present two contributions: (1) we develop a novel way of mapping traffic and topology data to a demand and cost model; and (2) we fit this model on three large real-world networks: an European transit ISP, a content distribution network, and an academic research network, and run counterfactuals to evaluate the effects of different pricing strategies on both the ISP profit and the consumer surplus. We highlight three core findings. First, ISPs gain most of the profits with only three or four pricing tiers and likely have little incentive to increase granularity of pricing even further. Second, we show that consumer surplus follows closely, if not precisely, the increases in ISP profit with more pricing tiers. Finally, the common ISP practice of structuring tiered contracts according to the cost of carrying the traffic flows (e.g., offering a discount for traffic that is local) can be suboptimal and that dividing contracts based on both traffic demand and the cost of carrying it into only three or four tiers yields near-optimal profit for the ISP.

cs.NI↗

arXiv2011

A Market for Air Traffic Flow Management

The two somewhat conflicting requirements of efficiency and fairness make ATFM an unsatisfactorily solved problem, despite its overwhelming importance. In this paper, we present an economics motivated solution that is based on the notion of a free market. Our contention is that in fact the airlines themselves are the best judge of how to achieve efficiency and our market-based solution gives them the ability to pay, at the going rate, to buy away the desired amount of delay on a per flight basis. The issue of fairness is simply finessed away by our solution -- whoever pays gets smaller delays. We show how our solution has the potential of enabling travelers from a large spectrum of affordability and punctuality requirements to achieve an end that is most desirable to them. Our market model is particularly simple, requiring only one parameter per flight from the airline company. Furthermore, we show that it admits a combinatorial, strongly polynomial algorithm for computing an equilibrium landing schedule and prices.

cs.CC↗

arXiv2010

Rational Convex Programs, Their Feasibility, and the Arrow-Debreu Nash Bargaining Game

Over the last decade, combinatorial algorithms have been obtained for exactly solving several nonlinear convex programs. We first provide a formal context to this activity by introducing the notion of {\em rational convex programs} -- this also enables us to identify a number of questions for further study. So far, such algorithms were obtained for total problems only. Our main contribution is developing the methodology for handling non-total problems, i.e., their associated convex programs may be infeasible for certain settings of the parameters. The specific problem we study pertains to a Nash bargaining game, called ADNB, which is derived from the linear case of the Arrow-Debreu market model. We reduce this game to computing an equilibrium in a new market model called {\em flexible budget market}, and we obtain primal-dual algorithms for determining feasibility, as well as giving a proof of infeasibility and finding an equilibrium. We give an application of our combinatorial algorithm for ADNB to an important "fair" throughput allocation problem on a wireless channel.

cs.GT↗

arXiv2010

Non-Separable, Quasiconcave Utilities are Easy -- in a Perfect Price Discrimination Market Model

Recent results, establishing evidence of intractability for such restrictive utility functions as additively separable, piecewise-linear and concave, under both Fisher and Arrow-Debreu market models, have prompted the question of whether we have failed to capture some essential elements of real markets, which seem to do a good job of finding prices that maintain parity between supply and demand. The main point of this paper is to show that even non-separable, quasiconcave utility functions can be handled efficiently in a suitably chosen, though natural, realistic and useful, market model; our model allows for perfect price discrimination. Our model supports unique equilibrium prices and, for the restriction to concave utilities, satisfies both welfare theorems.

cs.GT↗

arXiv2009

2-Player Nash and Nonsymmetric Bargaining Games: Algorithms and Structural Properties

The solution to a Nash or a nonsymmetric bargaining game is obtained by maximizing a concave function over a convex set, i.e., it is the solution to a convex program. We show that each 2-player game whose convex program has linear constraints, admits a rational solution and such a solution can be found in polynomial time using only an LP solver. If in addition, the game is succinct, i.e., the coefficients in its convex program are ``small'', then its solution can be found in strongly polynomial time. We also give a non-succinct linear game whose solution can be found in strongly polynomial time.

cs.GT↗