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The PrizeCollecting Generalized Steiner Tree Problem Via A New Approach Of PrimalDual Schema
"... In this paper we study the prizecollecting version of the Generalized Steiner Tree problem. To the best of our knowledge, there is no general combinatorial technique in approximation algorithms developed to study the prizecollecting versions of various problems. These problems are studied on a cas ..."
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Cited by 45 (13 self)
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In this paper we study the prizecollecting version of the Generalized Steiner Tree problem. To the best of our knowledge, there is no general combinatorial technique in approximation algorithms developed to study the prizecollecting versions of various problems. These problems are studied on a case by case basis by Bienstock et al. [5] by applying an LProunding technique which is not a combinatorial approach. The main contribution of this paper is to introduce a general combinatorial approach towards solving these problems through novel primaldual schema (without any need to solve an LP). We fuse the primaldual schema with Farkas lemma to obtain a combinatorial 3approximation algorithm for the PrizeCollecting Generalized Steiner Tree problem. Our work also inspires a combinatorial algorithm [12] for solving a special case of Kelly’s problem [21] of pricing edges. We also consider the kforest problem, a generalization of kMST and kSteiner tree, and we show that in spite of these problems for which there are constant factor approximation algorithms, the kforest problem is much harder to approximate. In particular, obtaining an approximation factor better than O(n 1/6−ε) for kforest requires substantially new ideas including improving the approximation factor O(n 1/3−ε) for the notorious densest ksubgraph problem. We note that kforest and prizecollecting version of Generalized Steiner Tree are closely related to each other, since the latter is the Lagrangian relaxation of the former.
Fair allocation without trade
 In Proceedings of the 11th International Joint Conference on Autonomous Agents and MultiAgent Systems (AAMAS
"... We consider the ageold problem of allocating items among different agents in a way that is efficient and fair. Two papers, by Dolev et al. and Ghodsi et al., have recently studied this problem in the context of computer systems. Both papers had similar models for agent preferences, but advocated di ..."
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Cited by 34 (0 self)
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We consider the ageold problem of allocating items among different agents in a way that is efficient and fair. Two papers, by Dolev et al. and Ghodsi et al., have recently studied this problem in the context of computer systems. Both papers had similar models for agent preferences, but advocated different notions of fairness. We formalize both fairness notions in economic terms, extending them to apply to a larger family of utilities. Noting that in settings with such utilities efficiency is easily achieved in multiple ways, we study notions of fairness as criteria for choosing between different efficient allocations. Our technical results are algorithms for finding fair allocations corresponding to two fairness notions: Regarding the notion suggested by Ghodsi et al., we present a polynomialtime algorithm that computes an allocation for a general class of fairness notions, in which their notion is included. For the other, suggested by Dolev et al., we show that a competitive market equilibrium achieves the desired notion of fairness, thereby obtaining a polynomialtime algorithm that computes such a fair allocation and solving the main open problem raised by Dolev et al.
A Revealed Preference Approach to Computational Complexity in Economics
, 2010
"... One of the main building blocks of economics is the theory of the consumer, which postulates that consumers are utility maximizing. However, from a computational perspective, this model is called into question because the task of utility maximization subject to a budget constraint is computationally ..."
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Cited by 14 (3 self)
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One of the main building blocks of economics is the theory of the consumer, which postulates that consumers are utility maximizing. However, from a computational perspective, this model is called into question because the task of utility maximization subject to a budget constraint is computationally hard in the worstcase under reasonable assumptions. In this paper, we study the empirical consequences of strengthening consumer choice theory to enforce that utilities are computationally easy to maximize. We prove the possibly surprising result that computational constraints have no empirical consequences whatsoever for consumer choice theory. That is, a data set is consistent with a utility maximizing consumer if and only if a data set is consistent with a utility maximizing consumer having a utility function that can be maximized in strongly polynomial time. Our result motivates a general approach for posing questions about the empirical content of computational constraints: the revealed preference approach to computational complexity. The approach complements the conventional worstcase view of computational complexity in important ways, and is methodologically close to mainstream economics.
New results on rationality and strongly polynomial solvability in eisenberggale markets
 In Proceedings of 2nd Workshop on Internet and Network Economics
, 2006
"... Abstract. We study the structure of EG[2], the class of EisenbergGale markets with two agents. We prove that all markets in this class are rational and they admit strongly polynomial algorithms whenever the polytope containing the set of feasible utilities of the two agents can be described via a c ..."
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Cited by 11 (10 self)
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Abstract. We study the structure of EG[2], the class of EisenbergGale markets with two agents. We prove that all markets in this class are rational and they admit strongly polynomial algorithms whenever the polytope containing the set of feasible utilities of the two agents can be described via a combinatorial LP. This helps resolve positively the status of two markets left as open problems by [JV]: the capacity allocation market in a directed graph with two sourcesink pairs and the network coding market in a directed network with two sources. Our algorithms for solving the corresponding nonlinear convex programs are fundamentally different from those obtained by [JV]; whereas they use the primaldual schema, we use a carefully constructed binary search. 1
How Profitable are Strategic Behaviors in a Market?
"... Abstract. It is common wisdom that individuals behave strategically in economic environments. We consider Fisher markets with Leontief utilities and study strategic behaviors of individual buyers in market equilibria. While simple examples illustrate that buyers do get larger utilities when behaving ..."
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Cited by 3 (1 self)
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Abstract. It is common wisdom that individuals behave strategically in economic environments. We consider Fisher markets with Leontief utilities and study strategic behaviors of individual buyers in market equilibria. While simple examples illustrate that buyers do get larger utilities when behaving strategically, we show that the benefits can be quite limited: We introduce the concept of incentive ratio to capture the extent to which utility can be increased by strategic behaviors of an individual, and show that the incentive ratio of Leontief markets is less than 2. We also reveal that the incentive ratios are insensitive to market sizes. Potentially, the concept incentive ratio can have applications in other strategic settings as well. 1
On Competitiveness in Uniform Utility Allocation Markets
"... In this paper, we study competitive markets a market is competitive if increasing the endowment of any one buyer does not increase the equilibrium utility of any other buyer. In the Fisher setting, competitive markets contain all markets with weak gross substitutability (WGS), a property which enab ..."
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Cited by 3 (2 self)
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In this paper, we study competitive markets a market is competitive if increasing the endowment of any one buyer does not increase the equilibrium utility of any other buyer. In the Fisher setting, competitive markets contain all markets with weak gross substitutability (WGS), a property which enable efficient algorithms for equilibrium computation. We show that every uniform utility allocation (UUA) market which is competitive, is a submodular utility allocation (SUA) market. Our result provides evidence for the existence of efficient algoritheorems for the class of competitive markets.
Rationality and Strongly Polynomial Solvability of EisenbergGale Markets with Two Agents
"... Inspired by the convex program of Eisenberg and Gale which captures Fisher markets with linear utilities, Jain and Vazirani [STOC, 2007] introduced the class of EisenbergGale (EG) markets. We study the structure of EG(2) markets, the class of EisenbergGale markets with two agents. We prove that al ..."
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Cited by 1 (1 self)
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Inspired by the convex program of Eisenberg and Gale which captures Fisher markets with linear utilities, Jain and Vazirani [STOC, 2007] introduced the class of EisenbergGale (EG) markets. We study the structure of EG(2) markets, the class of EisenbergGale markets with two agents. We prove that all markets in this class are rational, that is, they have rational equilibrium, and they admit strongly polynomial time algorithms whenever the polytope containing the set of feasible utilities of the two agents can be described via a combinatorial LP. This helps resolve positively the status of two markets left as open problems by Jain and Vazirani: the capacity allocation market in a directed graph with two sourcesink pairs and the network coding market in a directed network with two sources. Our algorithms for solving the corresponding nonlinear convex programs are fundamentally different from those obtained by Jain and Vazirani; whereas they use the primaldual schema, our main tool is binary search powered by the strong LPduality theorem. 1
Eisenberggale markets: Rationality, strongly polynomial time solvability and competition monotonicity
, 2006
"... We study the structure of EG[2], the class of EisenbergGale markets with two agents. We prove that all markets in this class are rational and they admit strongly polynomial algorithms whenever the polytope containing the set of feasible utilities of the two agents can be described via a combinatori ..."
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Cited by 1 (1 self)
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We study the structure of EG[2], the class of EisenbergGale markets with two agents. We prove that all markets in this class are rational and they admit strongly polynomial algorithms whenever the polytope containing the set of feasible utilities of the two agents can be described via a combinatorial LP. This helps resolve positively the status of two markets left as open problems by [JV]: the capacity allocation market in a directed graph with two sourcesink pairs and the network coding market in a directed network with two sources. Our algorithms for solving the corresponding nonlinear convex programs are fundamentally different from those obtained by [JV]; whereas they use the primaldual schema, we use a carefully constructed binary search. We also settle a third open problem of [JV], that of determining whether the notion of competition monotonicity characterizes the class of SUA markets within UUA markets. We give a positive resolution of this problem as well.
Fair Allocation Without Trade Draft Comment are welcome
"... We consider the ageold problem of allocating items among different agents in a way that is efficient and fair. Two papers, by Dolev et al. and Ghodsi et al., have recently studied this problem in the context of computer systems. Both papers had similar models for agent preferences, but advocated di ..."
Abstract
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We consider the ageold problem of allocating items among different agents in a way that is efficient and fair. Two papers, by Dolev et al. and Ghodsi et al., have recently studied this problem in the context of computer systems. Both papers had similar models for agent preferences, but advocated different notions of fairness. We formalize both fairness notions in economic terms, extending them to apply to a larger family of utilities. Noting that in settings with such utilities efficiency is easily achieved in multiple ways, we study notions of fairness as criteria for choosing between different efficient allocations. Our technical results are algorithms for finding fair allocations corresponding to two fairness notions: Regarding the notion suggested by Ghodsi et al., we present a polynomialtime algorithm that computes an allocation for a general class of fairness notions, in which their notion is included. For the other, suggested by Dolev et al., we show that a competitive market equilibrium achieves the desired notion of fairness, thereby obtaining a polynomialtime algorithm that computes such a fair allocation and solving the main open problem raised by Dolev et al. 1