A Primal-Dual Online Deterministic Algorithm for Matching with Delays

Bieńkowski, Marcin; Kraska, Artur; Liu, Hsiang-Hsuan; Schmidt, Paweł

doi:10.1007/978-3-030-04693-4_4

Cited by 14 publications

(9 citation statements)

References 39 publications

(47 reference statements)

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“…Bienkowski et al [14] proposed the first deterministic algorithm for the MBPMD problem on general metrics and achieved a competitive ratio of O m 2.46 , where m denotes the number of requests in the sequence. Later, the competitive ratio was improved to O(m) by using a primal-dual deterministic algorithm [13]. Azar and Jacob Fanani [5] combined the ideas of these papers to provide a deterministic algorithm which is O m log(2/3+ε) -competitive for the 2-MPMD and the MBPMD problems.…”

Section: Related Workmentioning

confidence: 99%

Online k-Way Matching with Delays and the H-Metric

Melnyk¹,

Wang²,

Wattenhofer³

2021

Preprint

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In this paper, we study k-Way Min-cost Perfect Matching with Delays -the k-MPMD problem. This problem considers a metric space with n nodes. Requests arrive at these nodes in an online fashion. The task is to match these requests into sets of exactly k, such that space and time cost of all matched requests are minimized. The notion of the space cost requires a definition of an underlying metric space that gives distances of subsets of k elements. For k > 2, the task of finding a suitable metric space is at the core of our problem: We show that for some known generalizations to k = 3 points, such as the 2-metric [23] and the D-metric [43], there exists no competitive randomized algorithm for the 3-MPMD problem. The G-metrics [39] are defined for 3 points and allows for a competitive algorithm for the 3-MPMD problem. For k > 3 points, there exist two generalizations of the G-metrics known as n-and K-metrics [4,31]. We show that neither the n-metrics nor the K-metrics can be used for the k-MPMD problem. On the positive side, we introduce the H-metrics, the first metrics to allow for a solution of the k-MPMD problem for all k. In order to devise an online algorithm for the k-MPMD problem on the H-metrics, we embed the H-metric into trees with an O(log n) distortion. Based on this embedding result, we extend the algorithm proposed by Azar et al. [7] and achieve a competitive ratio of O(log n) for the k-MPMD problem.

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Section: Related Workmentioning

confidence: 99%

Online k-Way Matching with Delays and the H-Metric

Melnyk¹,

Wang²,

Wattenhofer³

2021

Preprint

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“…Another metric optimization problem with delay is the problem of matching with delay [2,19,18,4,9,10]. For this problem, arbitrary delay functions are intractable, and thus the main line of work focuses on linear delay functions.…”

Section: Related Workmentioning

confidence: 99%

General Framework for Metric Optimization Problems with Delay or with Deadlines

Azar

Touitou

2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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In this paper, we present a framework used to construct and analyze algorithms for online optimization problems with deadlines or with delay over a metric space. Using this framework, we present algorithms for several di erent problems. We present an O(D 2 )-competitive deterministic algorithm for online multilevel aggregation with delay on a tree of depth D, an exponential improvement over the O(D 4 2 D )-competitive algorithm of Bienkowski et al. (ESA '16). We also present an O(log 2 n)competitive randomized algorithm for online service with delay over any general metric space of n points, improving upon the O(log 4 n)-competitive algorithm by Azar et al. (STOC '17).In addition, we present the problem of online facility location with deadlines. In this problem, requests arrive over time in a metric space, and need to be served until their deadlines by facilities that are opened momentarily for some cost. We also consider the problem of facility location with delay, in which the deadlines are replaced with arbitrary delay functions. For those problems, we present O(log 2 n)-competitive algorithms, with n the number of points in the metric space.The algorithmic framework we present includes techniques for the design of algorithms as well as techniques for their analysis. arXiv:1904.07131v1 [cs.DS] 15 Apr 20192. An O(log 2 n)-competitive randomized algorithm for online service with delay over a metric space with n points. This improves upon the O(log 4 n)-competitive randomized algorithm in [5].3. An O(log 2 n)-competitive randomized algorithm for online facility location with deadlines over a metric space with n points.4. An O(log 2 n)-competitive randomized algorithm for online facility location with delay over a metric space with n points.Our algorithms all share a common framework, which we present. The framework provides general structure to both the algorithm and its analysis.Such an improvement for the online multilevel aggregation problem is only known for the special case of deadlines, as given in [13].The algorithms for online facility location with deadlines and with delay can be easily extended to the case in which the cost of opening a facility is di erent for each point in the metric space. This changes the competitiveness of the algorithms to O(log 2 ∆ + log ∆ log n), where ∆ is the aspect ratio of the metric space. Our TechniquesAll of our algorithms are based on corresponding competitive algorithms for HSTs. The randomized algorithms for general metric spaces are obtained through randomized HST embedding. The O(D 2 )competitive deterministic algorithm for online multilevel aggregation with delay on a tree is based on decomposing the tree into a forest of HSTs. This decomposition is similar to that used in [13] for the case of deadlines.The framework -algorithm design. In designing algorithms for the problems over HSTs, we use a certain framework. In an algorithm designed using the framework, there is a counter for every node (in the case of facility location) or every edge (in the case of online m...

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“…Bienkowski et al also noted that the algorithm of [3] can be used to provide an O (n)-competitive deterministic algorithm for a general known metric space. Recently, Bienkowski et al [6] provided a new primal-dual deterministic algorithm for MPMD on general metrics, with a competitive-ratio of O (m), m being the number of requests.…”

Section: Introductionmentioning

confidence: 99%

“…Bienkowski et al improved this result in [6] by providing a new O(m)-competitive LP-based algorithm. Briefly, their algorithm maintains a primal relaxation of the matching problem and its dual (the programs evolve in time as more requests arrive).…”

Section: Introductionmentioning

confidence: 99%

Deterministic Min-Cost Matching with Delays

Azar

Jacob-Fanani

2020

Theory Comput Syst

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We consider the online Minimum-Cost Perfect Matching with Delays (MPMD) problem introduced by Emek et al. (STOC 2016), in which a general metric space is given, and requests are submitted in different times in this space by an adversary. The goal is to match requests, while minimizing the sum of distances between matched pairs in addition to the time intervals passed from the moment each request appeared until it is matched.In the online Minimum-Cost Bipartite Perfect Matching with Delays (MBPMD) problem introduced by Ashlagi et al. (APPROX/RANDOM 2017), each request is also associated with one of two classes, and requests can only be matched with requests of the other class.Previous algorithms for the problems mentioned above, include randomized O (log n)-competitive algorithms for known and finite metric spaces, n being the size of the metric space, and a deterministic O (m)-competitive algorithm, m being the number of requests.We introduce O m log( 3 2 +ǫ) -competitive deterministic algorithms for both problems and for any fixed ǫ > 0. In particular, for a small enough ǫ the competitive ratio becomes O m 0.59 . These are the first deterministic algorithms for the mentioned online matching problems, achieving a sub-linear competitive ratio. Our algorithms do not need to know the metric space in advance.

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A Primal-Dual Online Deterministic Algorithm for Matching with Delays

Cited by 14 publications

References 39 publications

Online k-Way Matching with Delays and the H-Metric

Online k-Way Matching with Delays and the H-Metric

General Framework for Metric Optimization Problems with Delay or with Deadlines

Deterministic Min-Cost Matching with Delays

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