2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS) 2016
DOI: 10.1109/focs.2016.24
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Online Algorithms for Covering and Packing Problems with Convex Objectives

Abstract: We present online algorithms for covering and packing problems with (non-linear) convex objectives. The convex covering problem is defined as: min x∈R n + f (x) s.t. Ax ≥ 1, where f : R n + → R+ is a monotone convex function, and A is an m × n matrix with non-negative entries. In the online version, a new row of the constraint matrix, representing a new covering constraint, is revealed in each step and the algorithm is required to maintain a feasible and monotonically non-decreasing assignment x over time. We … Show more

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Cited by 43 publications
(57 citation statements)
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References 29 publications
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“…In that case, we can apply the Pegasos [80] algorithm, which yields the following result. Note that we focus on NUM with a linear objective, but LOCO is not limited to linear objectives and Theorem 3 can be applied to NUM with a general convex objective function,for example, using the algorithm in [6].…”
Section: Resultsmentioning
confidence: 99%
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“…In that case, we can apply the Pegasos [80] algorithm, which yields the following result. Note that we focus on NUM with a linear objective, but LOCO is not limited to linear objectives and Theorem 3 can be applied to NUM with a general convex objective function,for example, using the algorithm in [6].…”
Section: Resultsmentioning
confidence: 99%
“…Note that all the updates the local sequential algorithm makes at step i are based only on the values of x j ∀j ∈ V for which a i j 0 when y i arrives. Local sequential algorithms include most online algorithms, such as the algorithms in [14] for covering or packing linear programs; those in [6] for convex covering and packing problems with linear constraints; and in [24] for general convex conic covering problems. For example, NUM is a packing problem with linear constraints, and thus LOCO can be run with the algorithms of [14] or [6].…”
Section: A Local Optimization Frameworkmentioning
confidence: 99%
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“…Prior to this work, the most general results known for online problems with inventory constraints are for the class of problems termed online optimization with packing constraints, e.g., [6,8,11,16,17]. This stream of work developed an interesting algorithmic framework based on a primal-dual or multiplicative weights update approaches, which centers around maintaining a dual variable for each constraint, understood as a shadow (or pseudo) price for the constraint given the information thus far.…”
Section: Related Workmentioning
confidence: 99%
“…Logarithmic competitive ratios are quite common in prior work on approximation algorithms and online algorithms. Examples include: set cover (Lovász, 1975;Johnson, 1974), buy-at-bulk network design (Awerbuch and Azar, 1997), sparsest cut (Arora et al, 2009), the dial-a-ride problem (Charikar and Raghavachari, 1998), online k-server (Bansal et al, 2011), online packing/covering (Azar et al, 2016), online set cover (Alon et al, 2003), online network design (Umboh, 2015), and online paging (Fiat et al, 1991).…”
mentioning
confidence: 99%