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Distributed Fractional Packing and Maximum Weighted bMatching via TailRecursive Duality
"... Abstract. We present efficient distributed δapproximation algorithms for fractional packing and maximum weighted bmatching in hypergraphs, where δ is the maximum number of packing constraints in which a variable appears (for maximum weighted bmatching δ is the maximum edge degree — for graphs δ = ..."
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Abstract. We present efficient distributed δapproximation algorithms for fractional packing and maximum weighted bmatching in hypergraphs, where δ is the maximum number of packing constraints in which a variable appears (for maximum weighted bmatching δ is the maximum edge degree — for graphs δ = 2). (a) For δ = 2 the algorithm runs in O(log m) rounds in expectation and with high probability. (b) For general δ, the algorithm runs in O(log 2 m) rounds in expectation and with high probability. 1 Background and results Given a weight vector w ∈ IR m +, a coefficient matrix A ∈ IR n×m and a vector b ∈ IR n +, the fractional packing problem is to compute a vector x ∈ IR m + to maximize ∑m j=1 wjxj and at the same time meet all the constraints ∑m j=1 Aijxj ≤ bi (∀i = 1... n). We use δ to denote the maximum number of packing constraints in which a variable appears, that is, δ = maxj {i  Aij ̸ = 0}. In the centralized setting, fractional packing
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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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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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.
Monotone Closure of Relaxed Constraints in Submodular Optimization: Connections Between Minimization and Maximization: Extended Version
"... It is becoming increasingly evident that many machine learning problems may be reduced to some form of submodular optimization. Previous work addresses generic discrete approaches and specific relaxations. In this work, we take a generic view from a relaxation perspective. We show a relaxation fo ..."
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It is becoming increasingly evident that many machine learning problems may be reduced to some form of submodular optimization. Previous work addresses generic discrete approaches and specific relaxations. In this work, we take a generic view from a relaxation perspective. We show a relaxation formulation and simple rounding strategy that, based on the monotone closure of relaxed constraints, reveals analogies between minimization and maximization problems, and includes known results as special cases and extends to a wider range of settings. Our resulting approximation factors match the corresponding integrality gaps. The results in this paper complement, in a sense explained in the paper, related discrete gradient based methods [30], and are particularly useful given the ever increasing need for efficient submodular optimization methods in very largescale machine learning. For submodular maximization, a number of relaxation approaches have been proposed. A critical challenge for the practical applicability of these techniques, however, is the complexity of evaluating the multilinear extension. We show that this extension can be efficiently evaluated for a number of useful submodular functions, thus making these otherwise impractical algorithms viable for many realworld machine learning problems. 1