<i>k</i>-Servers with a Smile: Online Algorithms via Projections

Buchbinder, Niv; Gupta, Anupam; Molinaro, Marco; Joseph, Joseph; Naor,

doi:10.1137/1.9781611975482.7

Cited by 24 publications

(26 citation statements)

References 16 publications

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“…The recent results by Bubeck et al [8] and Buchbinder et al [9] for the k-server problem extend also to the (h, k)-server setting, implying that R X (h, k) = O(D • log(1/ϵ)) for HSTs of depth D when…”

Section: Weak Adversariesmentioning

confidence: 85%

Unbounded lower bound for k-server against weak adversaries

Bieńkowski

Byrka

Coester

et al. 2020

Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing

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We study the resource augmented version of the k-server problem, also known as the k-server problem against weak adversaries or the (h, k)-server problem. In this setting, an online algorithm using k servers is compared to an offline algorithm using h servers, where h ≤ k. For uniform metrics, it has been known since the seminal work of Sleator and Tarjan (1985) that for any ϵ > 0, the competitive ratio drops to a constant if k = (1 + ϵ) • h. This result was later generalized to weighted stars (Young 1994) and trees of bounded depth (Bansal et al. 2017). The main open problem for this setting is whether a similar phenomenon occurs on general metrics.We resolve this question negatively. With a simple recursive construction, we show that the competitive ratio is at least Ω(log log h), even as k → ∞. Our lower bound holds for both deterministic and randomized algorithms. It also disproves the existence of a competitive algorithm for the infinite server problem on general metrics.

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Section: Weak Adversariesmentioning

confidence: 85%

Unbounded lower bound for k-server against weak adversaries

Bieńkowski

Byrka

Coester

et al. 2020

Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing

View full text Add to dashboard Cite

show abstract

“…Works obtaining poly-logarithmic competitive ratio are more recent, starting with [BBMN15], and more recently, by [BCL + 18] and [Lee18]; this resulted in the first poly log k-competitive algorithm. ( [BGMN19] gives an alternate projection-based perspective on [BCL + 18].) A new LP relaxation was introduced by [BCL + 18], who then use a mirror descent strategy with a multi-level entropy regularizer to obtain the online dynamics.…”

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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“…Projection interpretation of [BGMN19]. In [BGMN19] it is shown that the original OnlineSet-Cover algorithm of [AAA + 09, BN09] is equivalent to the following. Maintain a fractional solution x.…”

Section: Connections To Other Algorithmsmentioning

confidence: 99%

Random Order Set Cover is as Easy as Offline

Gupta¹,

Kehne²,

Levin³

2021

Preprint

Self Cite

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We give a polynomial-time algorithm for OnlineSetCover with a competitive ratio of O(log mn) when the elements are revealed in random order, essentially matching the best possible offline bound of O(log n) and circumventing the Ω(log m log n) lower bound known in adversarial order. We also extend the result to solving pure covering IPs when constraints arrive in random order.The algorithm is a multiplicative-weights-based round-and-solve approach we call LearnOr-Cover. We maintain a coarse fractional solution that is neither feasible nor monotone increasing, but can nevertheless be rounded online to achieve the claimed guarantee (in the random order model). This gives a new offline algorithm for SetCover that performs a single pass through the elements, which may be of independent interest.

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k-Servers with a Smile: Online Algorithms via Projections

Cited by 24 publications

References 16 publications

Unbounded lower bound for k-server against weak adversaries

Unbounded lower bound for k-server against weak adversaries

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

Random Order Set Cover is as Easy as Offline

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