The Infinite Server Problem

Coester, Christian; Koutsoupias, Elias; Lazos, Philip

doi:10.1145/3456632

Cited by 3 publications

(4 citation statements)

References 28 publications

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“…Koutsoupias [18] showed a (2K − 1)-competitive algorithm for the K-server problem for every metric space, which is also K-competitive, in case the metric is the real line Bartal and Koutsoupias [4]. Other variants of the K-server problem include the (H, K)-server problem Bansal et al [3,2], the infinite server problem Coester et al [8] and the K-taxi problemFiat et al [14], Coester and Koutsoupias [9].…”

Section: Related Workmentioning

confidence: 99%

Reallocating Multiple Facilities on the Line

Fotakis

Kavouras

Kostopanagiotis

et al. 2019

Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence

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We study the multistage K-facility reallocation problem on the real line, where we maintain K facility locations over T stages, based on the stage-dependent locations of n agents. Each agent is connected to the nearest facility at each stage, and the facilities may move from one stage to another, to accommodate different agent locations. The objective is to minimize the connection cost of the agents plus the total moving cost of the facilities, over all stages. K-facility reallocation was introduced by de Keijzer and Wojtczak [10], where they mostly focused on the special case of a single facility. Using an LP-based approach, we present a polynomial time algorithm that computes the optimal solution for any number of facilities. We also consider online K-facility reallocation, where the algorithm becomes aware of agent locations in a stage-by-stage fashion. By exploiting an interesting connection to the classical K-server problem, we present a constant-competitive algorithm for K = 2 facilities. 1 consecutive timesteps. The stability of the solutions is modeled by introducing an additional moving cost (or switching cost), which has a different definition depending on the particular setting. Model and Motivation.In this work, we study the multistage K-facility reallocation problem on the real line, introduced by de Keijzer and Wojtczak [10]. In K-facility reallocation, K facilities are initially located at (x 0 1 , . . . , x 0 K ) on the real line. Facilities are meant to serve n agents for the next T days. At each day, each agent connects to the facility closest to its location and incurs a connection cost equal to this distance. The locations of the agents may change every day, thus we have to move facilities accordingly in order to reduce the connection cost. Naturally, moving a facility is not for free, but comes with the price of the distance that the facility was moved. Our goal is to specify the exact positions of the facilities at each day so that the total connection cost plus the total moving cost is minimized over all T days. In the online version of the problem, the positions of the agents at each stage t are revealed only after determining the locations of the facilities at stage t − 1.For a motivating example, consider a company willing to advertise its products. To this end, it organizes K advertising campaigns at different locations of a large city for the next T days. Based on planned events, weather forecasts, etc., the company estimates a population distribution over the locations of the city for each day. Then, the company decides to compute the best possible campaign reallocation with K campaigns over all days (see also [10] for more examples).de Keijzer and Wojtczak [10] fully characterized the optimal offline and online algorithms for the special case of a single facility and presented a dynamic programming algorithm for K ≥ 1 facilities with running time exponential in K. Despite the practical significance and the interesting theoretical properties of Kfacility reallocation, its computationa...

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Section: Related Workmentioning

confidence: 99%

Reallocating Multiple Facilities on the Line

Fotakis

Kavouras

Kostopanagiotis

et al. 2019

Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence

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“…In [CKL17], the infinite server problem (denoted ∞-server problem here) is introduced as a possible way to resolve the question on general metrics. This is the variant of the k-server problem where k ∞, and all infinitely many servers initially reside at the same point of the metric space.…”

Section: Weak Adversariesmentioning

confidence: 99%

“…Namely, Bar-Noy and Schieber [BE98, page 175] showed that D X (2, k) 2 for all k when X is the line metric. For large h, the lower bound on D X (h, k) was improved to 2.41 [BEJK19] using depth-2 trees and later to 3.14 [CKL17] by a reduction from the ∞-server problem. In the absence of any super-constant lower bounds, the (h, k)-server hypothesis continued to seem plausible.…”

Section: Weak Adversariesmentioning

confidence: 99%

Unbounded lower bound for k-server against weak adversaries

Bieńkowski¹,

Byrka²,

Coester³

et al. 2019

Preprint

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“…A lower bound of 𝑘 is known to hold in all metrics [42]. It is tight in many special cases [11,19,20,23,30,39,42,49], and it is within a factor of 2 − 1 𝑘 of the truth in all cases [38]. Thus, it was unexpected that in the randomized case the competitive ratio would vary widely among different metric spaces.…”

Section: Introductionmentioning

confidence: 99%

The Randomized 𝑘-Server Conjecture Is False!

Bubeck

Coester

Rabani

2023

Proceedings of the 55th Annual ACM Symposium on Theory of Computing

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We prove a few new lower bounds on the randomized competitive ratio for the 𝑘-server problem and other related problems, resolving some long-standing conjectures. In particular, for metrical task systems (MTS) we asympotically settle the competitive ratio and obtain the first improvement to an existential lower bound since the introduction of the model 35 years ago (in 1987). More concretely, we show:(1) There exist (𝑘 + 1)-point metric spaces in which the randomized competitive ratio for the 𝑘-server problem is Ω(log 2 𝑘). This refutes the folklore conjecture (which is known to hold in some families of metrics) that in all metric spaces with at least 𝑘 + 1 points, the competitive ratio is Θ(log 𝑘).(2) Consequently, there exist 𝑛-point metric spaces in which the randomized competitive ratio for MTS is Ω(log 2 𝑛). This matches the upper bound that holds for all metrics. The previously best existential lower bound was Ω(log 𝑛) (which was known to be tight for some families of metrics). ( 3) For all 𝑘 < 𝑛 ∈ N, for all 𝑛-point metric spaces the randomized 𝑘-server competitive ratio is at least Ω(log 𝑘), and consequently the randomized MTS competitive ratio is at least Ω(log 𝑛). These universal lower bounds are asymptotically tight. The previous bounds were Ω(log 𝑘/log log 𝑘) and Ω(log 𝑛/log log 𝑛), respectively. (4) The randomized competitive ratio for the 𝑤-set metrical service systems problem, and its equivalent width-𝑤 layered graph traversal problem, is Ω(𝑤 2 ). This slightly improves the previous lower bound and matches the recently discovered upper bound. (5) Our results imply improved lower bounds for other problems like 𝑘-taxi, distributed paging, and metric allocation.These lower bounds share a common thread, and other than the third bound, also a common construction.

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The Infinite Server Problem

Cited by 3 publications

References 28 publications

Reallocating Multiple Facilities on the Line

Reallocating Multiple Facilities on the Line

Unbounded lower bound for k-server against weak adversaries

The Randomized 𝑘-Server Conjecture Is False!

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