Results 1  10
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56
On approximately fair allocations of indivisible goods
 In ACM Conference on Electronic Commerce (EC
, 2004
"... We study the problem of fairly allocating a set of indivisible goods to a set of people from an algorithmic perspective. Fair division has been a central topic in the economic literature and several concepts of fairness have been suggested. The criterion that we focus on is the maximum envy between ..."
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Cited by 75 (2 self)
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We study the problem of fairly allocating a set of indivisible goods to a set of people from an algorithmic perspective. Fair division has been a central topic in the economic literature and several concepts of fairness have been suggested. The criterion that we focus on is the maximum envy between any pair of players. An allocation is called envyfree if every player prefers her own share than the share of any other player. When the goods are divisible or when there is sufficient amount of one divisible good, envyfree allocations always exist. In the presence of indivisibilities however this is not the case. We first show that when all goods are indivisible, there always exist allocations in which the envy is bounded by the maximum marginal utility and we present a simple polynomial time algorithm for computing such allocations. We further show that our algorithm can be applied to the continuous cakecutting model as well and obtain a procedure that produces ɛenvyfree allocations with a linear number of cuts. We then look at the optimization problem of finding an allocation with minimum possible envy. In the general case, there is no polynomial time algorithm (or even approximation algorithm) for the problem, unless P = NP. We consider natural special cases (e.g. additive utilities) which are closely related to a class of job scheduling problems. Polynomial time approximation algorithms as well as inapproximability results are obtained. Finally we investigate the problem of designing truthful mechanisms for producing allocations with bounded envy. 1
Efficiency and envyfreeness in fair division of indivisible goods: Logical representation and complexity
 In Proceedings of the 19th International Joint Conference on Artificial Intelligence (IJCAI2005
, 2005
"... and complexity ..."
Approximating Submodular Functions Everywhere
"... Submodular functions are a key concept in combinatorial optimization. Algorithms that involve submodular functions usually assume that they are given by a (value) oracle. Many interesting problems involving submodular functions can be solved using only polynomially many queries to the oracle, e.g., ..."
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Cited by 45 (4 self)
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Submodular functions are a key concept in combinatorial optimization. Algorithms that involve submodular functions usually assume that they are given by a (value) oracle. Many interesting problems involving submodular functions can be solved using only polynomially many queries to the oracle, e.g., exact minimization or approximate maximization. In this paper, we consider the problem of approximating a nonnegative, monotone, submodular function f on a ground set of size n everywhere, after only poly(n) oracle queries. Our main result is a deterministic algorithm that makes poly(n) oracle queries and derives a function ˆ f such that, for every set S, ˆ f(S) approximates f(S) within a factor α(n), where α(n) = √ n + 1 for rank functions of matroids and α(n) = O ( √ n log n) for general monotone submodular functions. Our result is based on approximately finding a maximum volume inscribed ellipsoid in a symmetrized polymatroid, and the analysis involves various properties of submodular functions and polymatroids. Our algorithm is tight up to logarithmic factors. Indeed, we show that no algorithm can achieve a factor better than Ω ( √ n / log n), even for rank functions of a matroid.
Robust submodular observation selection
, 2008
"... In many applications, one has to actively select among a set of expensive observations before making an informed decision. For example, in environmental monitoring, we want to select locations to measure in order to most effectively predict spatial phenomena. Often, we want to select observations wh ..."
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Cited by 44 (4 self)
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In many applications, one has to actively select among a set of expensive observations before making an informed decision. For example, in environmental monitoring, we want to select locations to measure in order to most effectively predict spatial phenomena. Often, we want to select observations which are robust against a number of possible objective functions. Examples include minimizing the maximum posterior variance in Gaussian Process regression, robust experimental design, and sensor placement for outbreak detection. In this paper, we present the Submodular Saturation algorithm, a simple and efficient algorithm with strong theoretical approximation guarantees for cases where the possible objective functions exhibit submodularity, an intuitive diminishing returns property. Moreover, we prove that better approximation algorithms do not exist unless NPcomplete problems admit efficient algorithms. We show how our algorithm can be extended to handle complex cost functions (incorporating nonunit observation cost or communication and path costs). We also show how the algorithm can be used to nearoptimally trade off expectedcase (e.g., the Mean Square Prediction Error in Gaussian Process regression) and worstcase (e.g., maximum predictive variance) performance. We show that many important machine learning problems fit our robust submodular observation selection formalism, and provide extensive empirical evaluation on several realworld problems. For Gaussian Process regression, our algorithm compares favorably with stateoftheart heuristics described in the geostatistics literature, while being simpler, faster and providing theoretical guarantees. For robust experimental design, our algorithm performs favorably compared to SDPbased algorithms.
Online stochastic packing applied to display ad allocation
 In ESA
, 2010
"... Inspired by online ad allocation, we study online stochastic packing linear programs from theoretical and practical standpoints. We first present a nearoptimal online algorithm for a general class of packing linear programs which model various online resource allocation problems including online va ..."
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Cited by 40 (4 self)
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Inspired by online ad allocation, we study online stochastic packing linear programs from theoretical and practical standpoints. We first present a nearoptimal online algorithm for a general class of packing linear programs which model various online resource allocation problems including online variants of routing, ad allocations, generalized assignment, and combinatorial auctions. As our main theoretical result, we prove that a simple primaldual trainingbased algorithm achieves a (1 − o(1))approximation guarantee in the random order stochastic model. This is a significant improvement over logarithmic or constantfactor approximations for the adversarial variants of the same problems (e.g. factor 1 − 1e for online ad allocation, and log(m) for online routing). We then focus on the online display ad allocation problem and study the efficiency and fairness of various trainingbased and online allocation algorithms on data sets collected from reallife display ad allocation system. Our experimental evaluation confirms the effectiveness of trainingbased primaldual algorithms on real data sets, and also indicate an intrinsic tradeoff between fairness and efficiency. 1
On allocations that maximize fairness
 Proceedings of Symposium on Discrete Algorithms (SODA
, 2008
"... We consider a problem known as the restricted assignment version of the maxmin allocation problem with indivisible goods. There are n items of various nonnegative values and m players. Every player is interested only in some of the items and has zero value for the other items. One has to distribute ..."
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Cited by 28 (2 self)
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We consider a problem known as the restricted assignment version of the maxmin allocation problem with indivisible goods. There are n items of various nonnegative values and m players. Every player is interested only in some of the items and has zero value for the other items. One has to distribute the items among the players in a way that maximizes a certain notion of fairness, namely, maximizes the minimum of the sum of values of items given to any player. Bansal and Sviridenko [STOC 2006] describe a linear programming relaxation for this problem, and present a rounding technique that recovers an allocation of value at least Ω(log log log m / log log m) of the optimum. We show that the value of this LP relaxation in fact approximates the optimum value to within a constant factor. Our proof is not constructive and does not by itself provide an efficient algorithm for finding an allocation that is within constant factors of optimal. 1
Dependent Randomized Rounding via Exchange Properties of Combinatorial Structures (Extended Abstract)
"... Abstract—We consider the problem of randomly rounding a fractional solution x in an integer polytope P ⊆ [0, 1] n to a vertex X of P, so that E[X] = x. Our goal is to achieve concentration properties for linear and submodular functions of the rounded solution. Such dependent rounding techniques, wi ..."
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Cited by 27 (2 self)
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Abstract—We consider the problem of randomly rounding a fractional solution x in an integer polytope P ⊆ [0, 1] n to a vertex X of P, so that E[X] = x. Our goal is to achieve concentration properties for linear and submodular functions of the rounded solution. Such dependent rounding techniques, with concentration bounds for linear functions, have been developed in the past for two polytopes: the assignment polytope (that is, bipartite matchings and bmatchings) [32], [19], [23], and more recently for the spanning tree polytope [2]. These schemes have led to a number of new algorithmic results. In this paper we describe a new swap rounding technique which can be applied in a variety of settings including matroids and matroid intersection, while providing Chernofftype concentration bounds for linear and submodular functions of the rounded solution. In addition to existing techniques based on negative correlation, we use a martingale argument to obtain an exponential tail estimate for monotone submodular functions. The rounding scheme explicitly exploits exchange properties of the underlying combinatorial structures, and highlights these properties as the basis for concentration bounds. Matroids and matroid intersection provide a unifying framework for several known applications [19], [23], [7], [22], [2] as well as new ones, and their generality allows a richer set of constraints to be incorporated easily. We give some illustrative examples, with a more comprehensive discussion deferred to a later version of the paper. I.
Maxmin allocation via degree lowerbounded arborescences
 In STOC ’09: Proceedings of the 41st annual ACM Symposium on Theory of computing
, 2009
"... We consider the problem of MaxMin allocation of indivisible goods. There are m items to be distributed among n players. Each player i has a nonnegative valuation pi j for an item j, and the goal is to allocate items to players so as to maximize the minimum total valuation received by each player. Th ..."
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Cited by 26 (1 self)
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We consider the problem of MaxMin allocation of indivisible goods. There are m items to be distributed among n players. Each player i has a nonnegative valuation pi j for an item j, and the goal is to allocate items to players so as to maximize the minimum total valuation received by each player. There is a large gap in our understanding of this problem. The best known positive result is an Õ ( √ n)approximation algorithm, while there is only a factor 2 hardness known. Better algorithms are known for the restricted assignment case where each item has exactly one nonzero value for the players. We study the effect of bounded degree for items: each item has a nonzero value for at most D players. We show that essentially the case D = 3 is equivalent to the general case, and give a 4approximation algorithm for D = 2. The current algorithmic results for MaxMin Allocation are based
New Constructive Aspects of the Lovász Local Lemma
"... The Lovász Local Lemma (LLL) is a powerful tool that gives sufficient conditions for avoiding all of a given set of “bad ” events, with positive probability. A series of results have provided algorithms to efficiently construct structures whose existence is nonconstructively guaranteed by the LLL, ..."
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Cited by 26 (4 self)
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The Lovász Local Lemma (LLL) is a powerful tool that gives sufficient conditions for avoiding all of a given set of “bad ” events, with positive probability. A series of results have provided algorithms to efficiently construct structures whose existence is nonconstructively guaranteed by the LLL, culminating in the recent breakthrough of Moser & Tardos. We show that the output distribution of the MoserTardos algorithm wellapproximates the conditional LLLdistribution – the distribution obtained by conditioning on all bad events being avoided. We show how a known bound on the probabilities of events in this distribution can be used for further probabilistic analysis and give new constructive and nonconstructive results. We also show that when an LLL application provides a small amount of slack, the number of resamplings of the
The efficiency of fair division
 In Proceedings of the 5th International Workshop on Internet and Network Economics (WINE
, 2009
"... Abstract. We study the impact of fairness on the efficiency of allocations. We consider three different notions of fairness, namely proportionality, envyfreeness, and equitability for allocations of divisible and indivisible goods and chores. We present a series of results on the price of fairness ..."
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Cited by 24 (0 self)
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Abstract. We study the impact of fairness on the efficiency of allocations. We consider three different notions of fairness, namely proportionality, envyfreeness, and equitability for allocations of divisible and indivisible goods and chores. We present a series of results on the price of fairness under the three different notions that quantify the efficiency loss in fair allocations compared to optimal ones. Most of our bounds are either exact or tight within constant factors. Our study is of an optimistic nature and aims to identify the potential of fairness in allocations. 1