Results 1  10
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11
Maximizing a Monotone Submodular Function subject to a Matroid Constraint
, 2008
"... Let f: 2 X → R+ be a monotone submodular set function, and let (X, I) be a matroid. We consider the problem maxS∈I f(S). It is known that the greedy algorithm yields a 1/2 approximation [14] for this problem. For certain special cases, e.g. max S≤k f(S), the greedy algorithm yields a (1 − 1/e)app ..."
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Cited by 22 (1 self)
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Let f: 2 X → R+ be a monotone submodular set function, and let (X, I) be a matroid. We consider the problem maxS∈I f(S). It is known that the greedy algorithm yields a 1/2 approximation [14] for this problem. For certain special cases, e.g. max S≤k f(S), the greedy algorithm yields a (1 − 1/e)approximation. It is known that this is optimal both in the value oracle model (where the only access to f is through a black box returning f(S) for a given set S) [28], and also for explicitly posed instances assuming P � = NP [10]. In this paper, we provide a randomized (1 − 1/e)approximation for any monotone submodular function and an arbitrary matroid. The algorithm works in the value oracle model. Our main tools are a variant of the pipage rounding technique of Ageev and Sviridenko [1], and a continuous greedy process that might be of independent interest. As a special case, our algorithm implies an optimal approximation for the Submodular Welfare Problem in the value oracle model [32]. As a second application, we show that the Generalized Assignment Problem (GAP) is also a special case; although the reduction requires X  to be exponential in the original problem size, we are able to achieve a (1 − 1/e − o(1))approximation for GAP, simplifying previously known algorithms. Additionally, the reduction enables us to obtain approximation algorithms for variants of GAP with more general constraints.
Price of Anarchy for Greedy Auctions
"... We study mechanisms for utilitarian combinatorial allocation problems, where agents are not assumed to be singleminded. This class of problems includes combinatorial auctions, multiunit auctions, unsplittable flow problems, and others. We focus on the problem of designing mechanisms that approximat ..."
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Cited by 18 (7 self)
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We study mechanisms for utilitarian combinatorial allocation problems, where agents are not assumed to be singleminded. This class of problems includes combinatorial auctions, multiunit auctions, unsplittable flow problems, and others. We focus on the problem of designing mechanisms that approximately optimize social welfare at every BayesNash equilibrium (BNE), which is the standard notion of equilibrium in settings of incomplete information. For a broad class of greedy approximation algorithms, we give a general blackbox reduction to deterministic mechanisms with almost no loss to the approximation ratio at any BNE. We also consider the special case of Nash equilibria in fullinformation games, where we obtain tightened results. This solution concept is closely related to the wellstudied price of anarchy. Furthermore, for a rich subclass of allocation problems, pure Nash equilibria are guaranteed to exist for our mechanisms. For many problems, the approximation factors we obtain at equilibrium improve upon the best known results for deterministic truthful mechanisms. In particular, we exhibit a simple deterministic mechanism for general combinatorial auctions that obtains an O ( √ m) approximation at every BNE. 1
Approximating Maximum Weight Matching in Nearlinear Time
"... Given a weighted graph, the maximum weight matching problem (MWM) is to find a set of vertexdisjoint edges with maximum weight. In the 1960s Edmonds showed that MWMs can be found in polynomial time. At present the fastest MWM algorithm, due to Gabow and Tarjan, runs in Õ(m √ n) time, where m and n ..."
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Cited by 10 (1 self)
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Given a weighted graph, the maximum weight matching problem (MWM) is to find a set of vertexdisjoint edges with maximum weight. In the 1960s Edmonds showed that MWMs can be found in polynomial time. At present the fastest MWM algorithm, due to Gabow and Tarjan, runs in Õ(m √ n) time, where m and n are the number of edges and vertices in the graph. Surprisingly, restricted versions of the problem, such as computing (1 − ɛ)approximate MWMs or finding maximum cardinality matchings, are not known to be much easier (on sparse graphs). The best algorithms for these problems also run in Õ(m √ n) time. In this paper we present the first nearlinear time algorithm for computing (1 − ɛ)approximate MWMs. Specifically, given an arbitrary realweighted graph and ɛ> 0, our algorithm computes such a matching in O(mɛ −2 log 3 n) time. The previous best approximate MWM algorithm with comparable running time could only guarantee a (2/3 − ɛ)approximate solution. In addition, we present a faster algorithm, running in O(m log n log ɛ −1) time, that computes a (3/4−ɛ)approximate MWM.
Matroid matching: the power of local search
 IN STOC
, 2010
"... We consider the classical matroid matching problem. Unweighted matroid matching for linearlyrepresented matroids was solved by Lovász, and the problem is known to be intractable for general matroids. We present a PTAS for unweighted matroid matching for general matroids. In contrast, we show that ..."
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Cited by 6 (0 self)
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We consider the classical matroid matching problem. Unweighted matroid matching for linearlyrepresented matroids was solved by Lovász, and the problem is known to be intractable for general matroids. We present a PTAS for unweighted matroid matching for general matroids. In contrast, we show that natural LP relaxations that have been studied have an Ω(n) integrality gap and, moreover, Ω(n) rounds of the SheraliAdams hierarchy are necessary to bring the gap down to a constant. More generally, for any fixed k ≥ 2 and ɛ> 0, we obtain a (k/2 + ɛ)approximation for matroid matching in kuniform hypergraphs, also known as the matroid kparity problem. As a consequence, we obtain a (k/2+ɛ)approximation for the problem of finding the maximumcardinality set in the intersection of k matroids. We also give a 3/2approximation for the weighted version of a special case of matroid matching, the matchoid problem.
Truthful Mechanisms via Greedy Iterative Packing
, 2009
"... An important research thread in algorithmic game theory studies the design of efficient truthful mechanisms that approximate the optimal social welfare. A fundamental question is whether an αapproximation algorithm translates into an αapproximate truthful mechanism. It is wellknown that plugging ..."
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Cited by 4 (0 self)
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An important research thread in algorithmic game theory studies the design of efficient truthful mechanisms that approximate the optimal social welfare. A fundamental question is whether an αapproximation algorithm translates into an αapproximate truthful mechanism. It is wellknown that plugging an αapproximation algorithm into the VCG technique may not yield a truthful mechanism. Hence, it is natural to investigate properties of approximation algorithms that enable their use in truthful mechanisms. The main contribution of this paper is to identify a useful and natural property of approximation algorithms, which we call loserindependence. Intuitively, a loserindependent algorithm does not change its outcome when the bid of a losing agent increases, unless that agent becomes a winner. We demonstrate that loserindependent algorithms can be employed as subprocedures in a greedy iterative packing approach while preserving monotonicity. A greedy iterative approach provides good approximation in the context of maximizing a nondecreasing submodular function subject to independence constraints. Our framework gives rise to truthful approximation mechanisms for various problems. Notably, some problems arise in online mechanism design.
Improved Approximations for kExchange Systems (Extended Abstract)
"... Submodular maximization and set systems play a major role in combinatorial optimization. It is long known that the greedy algorithm provides a 1/(k + 1)approximation for maximizing a monotone submodular function over a ksystem. For the special case of kmatroid intersection, a local search appro ..."
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Cited by 1 (0 self)
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Submodular maximization and set systems play a major role in combinatorial optimization. It is long known that the greedy algorithm provides a 1/(k + 1)approximation for maximizing a monotone submodular function over a ksystem. For the special case of kmatroid intersection, a local search approach was recently shown to provide an improved approximation of 1/(k + δ) for arbitrary δ> 0. Unfortunately, many fundamental optimization problems are represented by a ksystem which is not a kintersection. An interesting question is whether the local search approach can be extended to include such problems. We answer this question affirmatively. Motivated by the bmatching and kset packing problems, as well as the more general matroid kparity problem, we introduce a new class of set systems called kexchange systems, that includes kset packing, bmatching, matroid kparity in strongly base orderable matroids, and additional combinatorial optimization problems such as: independent set in (k + 1)claw free graphs, asymmetric TSP, job interval selection with identical lengths and frequency allocation on lines. We give a natural local search algorithm which improves upon the current greedy approximation, for this new class of independence systems. Unlike known local search algorithms for similar problems, we use counting arguments to bound the performance of our algorithm. Moreover, we consider additional objective functions and provide improved approximations for them as well. In the case of linear objective functions, we give a nonoblivious local search algorithm, that improves upon existing local search approaches for matroid kparity.
The Power of Uncertainty: Algorithmic Mechanism Design in Settings of Incomplete Information
, 2011
"... The field of algorithmic mechanism design is concerned with the design of computationally efficient algorithms for use when inputs are provided by rational agents, who may misreport their private values in order to strategically manipulate the algorithm for their own benefit. We revisit classic prob ..."
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The field of algorithmic mechanism design is concerned with the design of computationally efficient algorithms for use when inputs are provided by rational agents, who may misreport their private values in order to strategically manipulate the algorithm for their own benefit. We revisit classic problems in this field by considering settings of incomplete information, where the players ’ private values are drawn from publiclyknown distributions. Such Bayesian models of partial information are common in economics, but have been largely unexplored by the computer science community. In the first part of this thesis we show that, for a very broad class of singleparameter problems, any computationally efficient algorithm can be converted without loss into a mechanism that is truthful in the Bayesian sense of partial information. That is, we exhibit a transformation that generates mechanisms for which it is in each agent’s best (expected) interest to refrain from strategic manipulation. The problem of constructing mechanisms for use by rational agents therefore reduces to the design of approximation algorithms without consideration of gametheoretic issues. We furthermore prove that
Truthful Manytomany Assignment with Private Weights ⋆
"... Abstract. This paper is devoted to the study of truthful mechanisms without payment for the manytomany assignment problem. Given n agents and m tasks, a mechanism is truthful if no agent has an incentive to misreport her values on the tasks (agent ai reports a score wij for each task tj). The one ..."
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Abstract. This paper is devoted to the study of truthful mechanisms without payment for the manytomany assignment problem. Given n agents and m tasks, a mechanism is truthful if no agent has an incentive to misreport her values on the tasks (agent ai reports a score wij for each task tj). The onetoone version of this problem has already been studied by Dughmi and Ghosh [4] in a setting where the weights wij are public knowledge, and the agents only report the tasks they are able to perform. We study here the case where the weights are private data. We are interested in the best approximation ratios that can be achieved by a truthful mechanism. In particular, we investigate the problem under various assumptions on the way the agents can misreport the weights. Key words: Algorithmic game theory, truthful mechanism without payment, approximation algorithm, manytomany assignment problem. 1
Matroid Matching: the Power of Local Search [Extended Abstract]
"... We consider the classical matroid matching problem. Unweighted matroid matching for linear matroids was solved by Lovász, and the problem is known to be intractable for general matroids. We present a PTAS for unweighted matroid matching for general matroids. In contrast, we show that natural LP rela ..."
Abstract
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We consider the classical matroid matching problem. Unweighted matroid matching for linear matroids was solved by Lovász, and the problem is known to be intractable for general matroids. We present a PTAS for unweighted matroid matching for general matroids. In contrast, we show that natural LP relaxations have an Ω(n) integrality gap and moreover, Ω(n) rounds of the SheraliAdams hierarchy are necessary to bring the gap down to a constant. More generally, for any fixed k ≥ 2 and ɛ> 0, we obtain a (k/2 + ɛ)approximation for matroid matching in kuniform hypergraphs, also known as the matroid kparity problem. As a consequence, we obtain a (k/2 + ɛ)approximation for the problem of finding the maximumcardinality set in the intersection of k matroids. We have also designed a 3/2approximation for the weighted version of a special case of matroid matching, the matchoid problem.
Structured Prediction of Generalized Matching Graphs
"... A structured prediction approach is proposed for completing missing edges in a graph using partially observed connectivity between n nodes. Unlike previous approaches, edge predictions depend on the node attributes (features) as well as graph topology. To overcome unrealistic i.i.d. edge prediction ..."
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A structured prediction approach is proposed for completing missing edges in a graph using partially observed connectivity between n nodes. Unlike previous approaches, edge predictions depend on the node attributes (features) as well as graph topology. To overcome unrealistic i.i.d. edge prediction assumptions, the structured prediction framework is extended to an output space of directed subgraphs that satisfy indegree and outdegree constraints. An efficient cutting plane algorithm is provided which interleaves the estimation of an edge score function with exact inference of the maximum weight degreeconstrained subgraph. Experiments with social networks, proteinprotein interaction graphs and citation networks are shown.