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Submodular function maximization via the multilinear relaxation and contention resolution schemes
 IN ACM SYMPOSIUM ON THEORY OF COMPUTING
, 2011
"... We consider the problem of maximizing a nonnegative submodular set function f: 2 N → R+ over a ground set N subject to a variety of packing type constraints including (multiple) matroid constraints, knapsack constraints, and their intersections. In this paper we develop a general framework that all ..."
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Cited by 40 (2 self)
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We consider the problem of maximizing a nonnegative submodular set function f: 2 N → R+ over a ground set N subject to a variety of packing type constraints including (multiple) matroid constraints, knapsack constraints, and their intersections. In this paper we develop a general framework that allows us to derive a number of new results, in particular when f may be a nonmonotone function. Our algorithms are based on (approximately) solving the multilinear extension F of f [5] over a polytope P that represents the constraints, and then effectively rounding the fractional solution. Although this approach has been used quite successfully in some settings [6, 22, 24, 13, 3], it has been limited in some important ways. We overcome these limitations as follows. First, we give constant factor approximation algorithms to maximize
When LP is the Cure for Your Matching Woes: Improved Bounds for Stochastic Matchings (Extended Abstract)
"... Abstract. Consider a random graph model where each possible edge e is present independently with some probability pe. We are given these numbers pe, and want to build a large/heavy matching in the randomly generated graph. However, the only way we can find out whether an edge is present or not is to ..."
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Cited by 23 (5 self)
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Abstract. Consider a random graph model where each possible edge e is present independently with some probability pe. We are given these numbers pe, and want to build a large/heavy matching in the randomly generated graph. However, the only way we can find out whether an edge is present or not is to query it, and if the edge is indeed present in the graph, we are forced to add it to our matching. Further, each vertex i is allowed to be queried at most ti times. How should we adaptively query the edges to maximize the expected weight of the matching? We consider several matching problems in this general framework (some of which arise in kidney exchanges and online dating, and others arise in modeling online advertisements); we give LProunding based constantfactor approximation algorithms for these problems. Our main results are: • We give a 5.75approximation for weighted stochastic matching on general graphs, and a 5approximation on bipartite graphs. This answers an open question from [Chen et al. ICALP 09]. • Combining our LProunding algorithm with the natural greedy algorithm, we give an improved 3.88approximation for unweighted stochastic matching on general graphs and 3.51approximation on bipartite graphs. • We introduce a generalization of the stochastic online matching problem [Feldman et al. FOCS 09] that also models preferenceuncertainty and timeouts of buyers, and give a constant factor approximation algorithm. 1
A Unified Continuous Greedy Algorithm for Submodular Maximization
, 2011
"... The study of combinatorial problems with a submodular objective function has attracted much attention in recent years, and is partly motivated by the importance of such problems to economics, algorithmic game theory and combinatorial optimization. Classical works on these problems are mostly combin ..."
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Cited by 21 (4 self)
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The study of combinatorial problems with a submodular objective function has attracted much attention in recent years, and is partly motivated by the importance of such problems to economics, algorithmic game theory and combinatorial optimization. Classical works on these problems are mostly combinatorial in nature. Recently, however, many results based on continuous algorithmic tools have emerged. The main bottleneck of such continuous techniques is how to approximately solve a nonconvex relaxation for the submodular problem at hand. Thus, the efficient computation of better fractional solutions immediately implies improved approximations for numerous applications. A simple and elegant method, called “continuous greedy”, successfully tackles this issue for monotone submodular objective functions, however,
Approximability of sparse integer programs
 In Proc. 17th ESA
, 2009
"... The main focus of this paper is a pair of new approximation algorithms for sparse integer programs. First, for covering integer programs {min cx: Ax ≥ b,0 ≤ x ≤ d} where A has at most k nonzeroes per row, we give a kapproximation algorithm. (We assume A, b, c, d are nonnegative.) For any k ≥ 2 and ..."
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Cited by 9 (1 self)
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The main focus of this paper is a pair of new approximation algorithms for sparse integer programs. First, for covering integer programs {min cx: Ax ≥ b,0 ≤ x ≤ d} where A has at most k nonzeroes per row, we give a kapproximation algorithm. (We assume A, b, c, d are nonnegative.) For any k ≥ 2 and ǫ> 0, if P = NP this ratio cannot be improved to k − 1 − ǫ, and under the unique games conjecture this ratio cannot be improved to k − ǫ. One key idea is to replace individual constraints by others that have better rounding properties but the same nonnegative integral solutions; another critical ingredient is knapsackcover inequalities. Second, for packing integer programs {max cx: Ax ≤ b,0 ≤ x ≤ d} where A has at most k nonzeroes per column, we give a 2 k k 2approximation algorithm. This is the first polynomialtime approximation algorithm for this problem with approximation ratio depending only on k, for any k> 1. Our approach starts from iterated LP relaxation, and then uses probabilistic and greedy methods to recover a feasible solution. Note added after publication: This version includes subsequent developments: a O(k 2) approximation for the latter problem using the iterated rounding framework, and several literature reference updates including a O(k)approximation for the same problem by Bansal et al.
On the Complexity of PrivacyPreserving Complex Event Processing ABSTRACT
"... Complex Event Processing (CEP) Systems are stream processing systems that monitor incoming event streams in search of userspecified event patterns. While CEP systems have been adopted in a variety of applications, the privacy implications of event pattern reporting mechanisms have yet to be studied ..."
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Cited by 6 (1 self)
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Complex Event Processing (CEP) Systems are stream processing systems that monitor incoming event streams in search of userspecified event patterns. While CEP systems have been adopted in a variety of applications, the privacy implications of event pattern reporting mechanisms have yet to be studied — a stark contrast to the significant amount of attention that has been devoted to privacy for relational systems. In this paper we present a privacy problem that arises when the system must support desired patterns (those that should be reported if detected) and private patterns (those that should not be revealed). We formalize this problem, which we term privacypreserving, utility maximizing CEP (PPCEP), and analyze its complexity under various assumptions. Our results show that this is a rich problem to study and shed some light on the difficulty of developing algorithms that preserve utility without compromising privacy. 1.
Precoder design for physical layer multicasting
 IEEE Transactions on Signal Processing
, 2012
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Efficient Submodular Function Maximization under Linear Packing Constraints
"... We study the problem of maximizing a monotone submodular set function subject to linear packing constraints. An instance of this problem consists of a matrix A ∈ [0, 1] m×n, a vector b ∈ [1, ∞) m, and a monotone submodular set function f: 2 [n] → R+. The objective is to find a set S that maximizes ..."
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Cited by 4 (0 self)
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We study the problem of maximizing a monotone submodular set function subject to linear packing constraints. An instance of this problem consists of a matrix A ∈ [0, 1] m×n, a vector b ∈ [1, ∞) m, and a monotone submodular set function f: 2 [n] → R+. The objective is to find a set S that maximizes f(S) subject to AxS ≤ b. Here, xS stands for the characteristic vector of the set S. A wellstudied special case of this problem is when the objective function f is linear. This special case captures the class of packing integer programs. Our main contribution is an efficient combinatorial algorithm that achieves an approximation ratio of Ω(1/m 1/W), where W = min{bi/Aij: Aij> 0} is the width of the packing constraints. This result matches the best known performance guarantee for the linear case. One immediate corollary of this result is that the algorithm under consideration achieves constant factor approximation when the number of constraints is constant or when the width of the packing constraints is sufficiently large. This motivates us to study the large width setting, trying to determine its exact approximability. We develop an algorithm that has an approximation ratio of (1 − ɛ)(1 − 1/e) when W = Ω(ln m/ɛ 2). This result (almost) matches the theoretical lower bound of 1−1/e, which already holds for maximizing a monotone submodular function subject to a cardinality constraint.
Distributed Algorithms for Covering, Packing and Maximum Weighted Matching
"... This paper gives polylogarithmicround, distributed δapproximation algorithms for covering problems with submodular cost and monotone covering constraints (Submodularcost Covering). The approximation ratio δ is the maximum number of variables in any constraint. Special cases include Covering Mix ..."
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Cited by 4 (1 self)
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This paper gives polylogarithmicround, distributed δapproximation algorithms for covering problems with submodular cost and monotone covering constraints (Submodularcost Covering). The approximation ratio δ is the maximum number of variables in any constraint. Special cases include Covering Mixed Integer Linear Programs (CMIP), and Weighted Vertex Cover (with δ = 2). Via duality, the paper also gives polylogarithmicround, distributed δapproximation algorithms for Fractional Packing linear programs (where δ is the maximum number of constraints in which any variable occurs), and for Max Weighted cMatching in hypergraphs (where δ is the maximum size of any of the hyperedges; for graphs δ = 2). The paper also gives parallel (RNC) 2approximation algorithms for CMIP with two variables per constraint and Weighted Vertex Cover. The algorithms are randomized. All of the approximation ratios exactly match those of comparable centralized algorithms.
Approximations for Monotone and Nonmonotone Submodular Maximization with Knapsack Constraints
"... Submodular maximization generalizes many fundamental problems in discrete optimization, including MaxCut in directed/undirected graphs, maximum coverage, maximum facility location and marketing over social networks. In this paper we consider the problem of maximizing any submodular function subject ..."
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Cited by 4 (0 self)
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Submodular maximization generalizes many fundamental problems in discrete optimization, including MaxCut in directed/undirected graphs, maximum coverage, maximum facility location and marketing over social networks. In this paper we consider the problem of maximizing any submodular function subject to d knapsack constraints, where d is a fixed constant. We establish a strong relation between the discrete problem and its continuous relaxation, obtained through extension by expectation of the submodular function. Formally, we show that, for any nonnegative submodular function, an αapproximation algorithm for the continuous relaxation implies a randomized (α−ε)approximation algorithm for the discrete problem. We use this relation to obtain an (e −1 −ε)approximation for the problem, and a nearly optimal (1 − e −1 − ε)−approximation ratio for the monotone case, for any ε> 0. We further show that the probabilistic domain defined by a continuous solution can be reduced to yield a polynomial size domain, given an oracle for the extension by expectation. This leads to a deterministic version of our technique.
A stochastic probing problem with applications
 In Proc. of 16th IPCO. Forthcoming
, 2013
"... Abstract. We study a general stochastic probing problem defined on a universe V, where each element e ∈ V is “active ” independently with probability pe. Elementshaveweights{we: e ∈ V} and the goal is to maximize the weight of a chosen subset S of active elements. However, we are given only the pe v ..."
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Cited by 3 (1 self)
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Abstract. We study a general stochastic probing problem defined on a universe V, where each element e ∈ V is “active ” independently with probability pe. Elementshaveweights{we: e ∈ V} and the goal is to maximize the weight of a chosen subset S of active elements. However, we are given only the pe values—to determine whether or not an element e is active, our algorithm must probe e. Ifelementeis probed and happens to be active, then e must irrevocably be added to the chosen set S; ifeis not active then it is not included in S. Moreover, the following conditions must hold in every random instantiation: – the set Q of probed elements satisfy an “outer ” packing constraint, – the set S of chosen elements satisfy an “inner ” packing constraint. The kinds of packing constraints we consider are intersections of matroids and knapsacks. Our results provide a simple and unified view of results in stochastic matching [1, 2] and Bayesian mechanism design [3], and can also handle more general constraints. As an application, we obtain the first polynomialtime Ω(1/k)approximate “Sequential Posted Price Mechanism ” under kmatroid intersection feasibility constraints, improving on prior work [3–5]. 1