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On kColumn Sparse Packing Programs
, 2009
"... We consider the class of packing integer programs (PIPs) that are column sparse, i.e. there is a specified upper bound k on the number of constraints that each variable appears in. We give an ek+o(k)approximation algorithm for kcolumn sparse PIPs, improving on recent results of k2 · 2k [14] and O(k ..."
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We consider the class of packing integer programs (PIPs) that are column sparse, i.e. there is a specified upper bound k on the number of constraints that each variable appears in. We give an ek+o(k)approximation algorithm for kcolumn sparse PIPs, improving on recent results of k2 · 2k [14] and O(k2) [3, 5]. We also show that the integrality gap of our linear programming relaxation is at least 2k − 1; it is known that kcolumn sparse PIPs are Ω(k log k)hard to approximate [8]. We also extend our result (at the loss of a small constant factor) to the more general case of maximizing a submodular objective over kcolumn sparse packing constraints.
Unsplittable flow in paths and trees and columnrestricted packing integer programs
 IN PROCEEDINGS, INTERNATIONAL WORKSHOP ON APPROXIMATION ALGORITHMS FOR COMBINATORIAL OPTIMIZATION PROBLEMS
, 2009
"... We consider the unsplittable flow problem (UFP) and the closely related columnrestricted packing integer programs (CPIPs). In UFP we are given an edgecapacitated graph G = (V, E) and k request pairs R1,..., Rk, where each Ri consists of a sourcedestination pair (si, ti), a demand di and a weigh ..."
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Cited by 11 (0 self)
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We consider the unsplittable flow problem (UFP) and the closely related columnrestricted packing integer programs (CPIPs). In UFP we are given an edgecapacitated graph G = (V, E) and k request pairs R1,..., Rk, where each Ri consists of a sourcedestination pair (si, ti), a demand di and a weight wi. The goal is to find a maximum weight subset of requests that can be routed unsplittably in G. Most previous work on UFP has focused on the nobottleneck case in which the maximum demand of the requests is at most the smallest edge capacity. Inspired by the recent work of Bansal et al. [3] on UFP on a path without the above assumption, we consider UFP on paths as well as trees. We give a simple O(log n) approximation for UFP on trees when all weights are identical; this yields an O(log 2 n) approximation for the weighted case. These are the first nontrivial approximations for UFP on trees. We develop an LP relaxation for UFP on paths that has an integrality gap of O(log 2 n); previously there was no relaxation with o(n) gap. We also consider UFP in general graphs and CPIPs without the nobottleneck assumption and obtain new and useful results.
Randomized Pipage Rounding for Matroid Polytopes and Applications
, 2009
"... We present concentration bounds for linear functions of random variables arising from the pipage rounding procedure on matroid polytopes. As an application, we give a (1 − 1/e − ɛ)approximation algorithm for the problem of maximizing a monotone submodular function subject to 1 matroid and k linear ..."
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Cited by 9 (1 self)
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We present concentration bounds for linear functions of random variables arising from the pipage rounding procedure on matroid polytopes. As an application, we give a (1 − 1/e − ɛ)approximation algorithm for the problem of maximizing a monotone submodular function subject to 1 matroid and k linear constraints, for any constant k ≥ 1 and ɛ> 0. This generalizes the result for k linear constraints by Kulik et al. [11]. We also give the same result for a superconstant number k of ”loose ” linear constraints, where the righthand side dominates the matrix entries by an Ω(ɛ −2 log k) factor. As another application, we present a general result on minimax packing problems that involve a matroid base constraint. An example is the multipath routing problem with integer demands for pairs of vertices; the goal is to minimize congestion. We give an O(log m / log log m)approximation for the general problem min{λ: ∃x ∈ {0, 1} N, x ∈ B(M), Ax ≤ λb} where m is the number of packing constraints.
On the Complexity of PrivacyPreserving Complex Event Processing
"... 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 7 (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.
Greedy ΔApproximation Algorithm for Covering with Arbitrary Constraints and Submodular Cost
 ALGORITHMICA
, 2012
"... This paper describes a simple greedy Δapproximation algorithm for any covering problem whose objective function is submodular and nondecreasing, and whose feasible region can be expressed as the intersection of arbitrary (closed upwards) covering constraints, each of which constrains at most Δ var ..."
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Cited by 6 (1 self)
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This paper describes a simple greedy Δapproximation algorithm for any covering problem whose objective function is submodular and nondecreasing, and whose feasible region can be expressed as the intersection of arbitrary (closed upwards) covering constraints, each of which constrains at most Δ variables of the problem. (A simple example is VERTEX COVER, with Δ = 2.) The algorithm generalizes previous approximation algorithms for fundamental covering problems and online paging and caching problems.
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.
Solving Packing Integer Programs via Randomized Rounding with Alterations
 THEORY OF COMPUTING
, 2012
"... We give new approximation algorithms for packing integer programs (PIPs) by employing the method of randomized rounding combined with alterations. Our first result is a simpler approximation algorithm for general PIPs which matches the best known bounds, and which admits an efficient parallel imple ..."
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Cited by 2 (0 self)
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We give new approximation algorithms for packing integer programs (PIPs) by employing the method of randomized rounding combined with alterations. Our first result is a simpler approximation algorithm for general PIPs which matches the best known bounds, and which admits an efficient parallel implementation. We also extend these results to a multicriteria version of PIPs. Our second result is for the class of packing integer programs (PIPs) that are column sparse, i.e., where there is a specified upper bound k on the number of constraints that each variable appears in. We give an (ek + o(k))approximation algorithm for kcolumn sparse PIPs, improving over previously known O(k2)approximation ratios. We also generalize our result to the case of maximizing nonnegative monotone submodular) functions over e2k e−1 kcolumn sparse packing constraints, and obtain an + o(k)approximation algorithm. In obtaining this result, we prove a new property of submodular functions that generalizes the fractional subadditivity property, which might be of independent interest.
Approximation Schemes for Deal Splitting and Covering Integer Programs with Multiplicity Constraints
"... We consider the problem of splitting an order for R goods, R ≥ 1, among a set of sellers, each having bounded amounts of the goods, so as to minimize the total cost of the deal. In deal splitting with packages (DSP), the sellers offer packages containing combinations of the goods; in deal splitting ..."
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We consider the problem of splitting an order for R goods, R ≥ 1, among a set of sellers, each having bounded amounts of the goods, so as to minimize the total cost of the deal. In deal splitting with packages (DSP), the sellers offer packages containing combinations of the goods; in deal splitting with price tables (DST), the buyer can generate such combinations using price tables. Our problems, which often occur in online reverse auctions, generalize covering integer programs with multiplicity constraints (CIP), where we must fill up an Rdimensional bin by selecting (with bounded number of repetitions) from a set of Rdimensional items, such that the overall cost is minimized. Thus, both DSP and DST are NPhard, already for a single good, and hard to approximate for arbitrary number of goods. In this paper we focus on finding efficient approximations for DSP and DST instances where the number of goods is some fixed constant. In particular, we develop polynomial time approximation schemes (PTAS) for several subclasses of instances of practical interest. Our results include a PTAS for CIP in fixed dimension, and a more efficient (combinatorial) scheme for CIP∞, where the multiplicity constraints are omitted. Our approximation scheme for CIP ∞ is based on a nontrivial application of the fast scheme for the fractional covering