Approximating Sparse Covering Integer Programs Online

Gupta, Anupam; Nagarajan, Viswanath

doi:10.1007/978-3-642-31594-7_37

Cited by 16 publications

(27 citation statements)

References 17 publications

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“…Prior to this work, analogous results for online covering/packing were only known for linear objectives. Competitive ratios of O(log n) for covering and O(log(ρd)) for packing were obtained in [17], the covering ratio being subsequently improved to O(log d) in [18]. For linear objectives, Theorem 1 obtains ratios of O(log d) for covering and O(log(ρd)) for packing, matching the best bounds.…”

Section: B Our Resultsmentioning

confidence: 83%

“…Our algorithm for convex covering and packing (Theorem 1) is based on a clean unified approach, that significantly simplifies even the well-studied linear case [18]. The value of the primal variables grows exponentially proportional both to their current value and their coefficient in the constraint, but divided by the gradient of f at the current solution.…”

Section: Our Techniquesmentioning

confidence: 99%

“…We give here a primal-dual algorithm for the online convex packing/covering problem, and prove Theorem 1. The algorithm is presented in Figure III and is very simple; in fact, even simpler than the optimal algorithm for the linear case [18].…”

Section: Online Covering Frameworkmentioning

confidence: 99%

“…In the online setting, generic linear programming solvers cannot be used for even obtaining a fractional solution, since constraints are presented online. Hence, new algorithms have been developed for solving linear programs online [17], [18].…”

Section: Introductionmentioning

confidence: 99%

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Online Algorithms for Covering and Packing Problems with Convex Objectives

Azar

Buchbinder

Chan

et al. 2016

2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)

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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 also consider a convex packing problem defined as: max y∈R m + m j=1 yj − g(A T y), where g : R n + → R+ is a monotone convex function. In the online version, each variable yj arrives online and the algorithm must decide the value of yj on its arrival. This represents the Fenchel dual of the convex covering program, when g is the convex conjugate of f. We use a primal-dual approach to give online algorithms for these generic problems, and use them to simplify, unify, and improve upon previous results for several applications.

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Section: B Our Resultsmentioning

confidence: 83%

Section: Our Techniquesmentioning

confidence: 99%

Section: Online Covering Frameworkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Online Algorithms for Covering and Packing Problems with Convex Objectives

Azar

Buchbinder

Chan

et al. 2016

2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)

Self Cite

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“…Algorithms for the online set cover problem were first given by [AAA + 09]: this led to the general primal-dual approach for covering linear programs (and sparse set-cover instances) [BN09], and to sparse CIPs [GN14]. Our algorithm also uses a similar primal-dual approach for the local LPs defined at each node of the tree; we also need to crucially use the sparsity properties of the corresponding set-cover-like constraints.…”

Section: Related Workmentioning

confidence: 99%

A Hitting Set Relaxation for $k$-Server and an Extension to Time-Windows

Gupta¹,

Kumar²,

Panigrahi³

2021

Preprint

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We study the k-server problem with time-windows. In this problem, each request i arrives at some point v i of an n-point metric space at time b i and comes with a deadline e i . One of the k servers must be moved to v i at some time in the interval [b i , e i ] to satisfy this request. We give an online algorithm for this problem with a competitive ratio of poly log(n, ∆), where ∆ is the aspect ratio of the metric space. Prior to our work, the best competitive ratio known for this problem was O(k poly log(n)) given by Azar et al. (STOC 2017).Our algorithm is based on a new covering linear program relaxation for k-server on HSTs. This LP naturally corresponds to the min-cost flow formulation of k-server, and easily extends to the case of time-windows. We give an online algorithm for obtaining a feasible fractional solution for this LP, and a primal dual analysis framework for accounting the cost of the solution. Together, they yield a new k-server algorithm with poly-logarithmic competitive ratio, and extend to the time-windows case as well. Our principal technical contribution lies in thinking of the covering LP as yielding a truncated covering LP at each internal node of the tree, which allows us to keep account of server movements across subtrees. We hope that this LP relaxation and the algorithm/analysis will be a useful tool for addressing k-server and related problems.

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Online k-Taxi via Double Coverage and Time-Reverse Primal-Dual

Buchbinder

Coester

Naor

2021

Integer Programming and Combinatorial Optimization

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We consider the online k-taxi problem, a generalization of the k-server problem, in which k servers are located in a metric space. A sequence of requests is revealed one by one, where each request is a pair of two points, representing the start and destination of a travel request by a passenger. The goal is to serve all requests while minimizing the distance traveled without carrying a passenger. We show that the classic Double Coverage algorithm has competitive ratio 2 k − 1 on HSTs, matching a recent lower bound for deterministic algorithms. For bounded depth HSTs, the competitive ratio turns out to be much better and we obtain tight bounds. When the depth is d ≪ k, these bounds are approximately k d /d!. By standard embedding results, we obtain a randomized algorithm for arbitrary n-point metrics with (polynomial) competitive ratio O(k c 1/c log n), where is the aspect ratio and c ≥ 1 is an arbitrary positive integer constant. The previous known bound was O(2 k log n). For general (weighted) tree metrics, we prove the competitive ratio of Double Coverage to be (k d ) for any fixed depth d, and in contrast to HSTs it is not bounded by 2 k − 1. We obtain our results by a dual fitting analysis where the dual solution is constructed step-by-step backwards in time. Unlike the forward-time approach typical of online primal-dual analyses, this allows us to combine information from both the past and the future when An extended abstract of this article appeared in IPCO 2021.

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Approximating Sparse Covering Integer Programs Online

Cited by 16 publications

References 17 publications

Online Algorithms for Covering and Packing Problems with Convex Objectives

Online Algorithms for Covering and Packing Problems with Convex Objectives

A Hitting Set Relaxation for $k$-Server and an Extension to Time-Windows

Online k-Taxi via Double Coverage and Time-Reverse Primal-Dual

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