A Collection of Lower Bounds for Online Matching on the Line

Antoniadis, Antonios; Fischer, Carsten; Tönnis, Andreas

doi:10.1007/978-3-319-77404-6_5

Cited by 16 publications

(16 citation statements)

References 17 publications

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“…Despite these advancements, there is still a big gap between the best known upper and lower bounds for the problem. Furthermore, Antoniadis, Fischer, and Tönnis [2] have shown a lower bound of Ω(log n) for a restricted class of algorithms. Since this class contains all the deterministic algorithms found in the literature, it would be interesting to try to either develop an algorithm that does not belong to this class, or extend their construction to show a lower bound for an even wider class of algorithms.…”

Section: Discussionmentioning

confidence: 99%

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A $${o}\mathopen {}\left( n\right) \mathclose {}$$-Competitive Deterministic Algorithm for Online Matching on a Line

et al. 2019

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Online matching on a line involves matching an online stream of requests to a given set of servers, all in the real line, with the objective of minimizing the sum of the distances between matched server-request pairs. The best previously known upper and lower bounds on the optimal deterministic competitive ratio are linear in the number of requests, and constant, respectively. We show that online matching on a line is essentially equivalent to a particular search problem, which we call k-lost-cows. We then obtain the first deterministic sub-linearly competitive algorithm for online matching on a line by giving such an algorithm for the k-lost-cows problem.

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Section: Discussionmentioning

confidence: 99%

“…We refer to this strategy as the (1 + )-cow algorithm. 0 −1 1 + −(1 + ) 2 (1 + ) 3 Fig. 3 Representation of the (1 + )-cow algorithm.…”

Section: Discussionmentioning

confidence: 99%

A $${o}\mathopen {}\left( n\right) \mathclose {}$$-Competitive Deterministic Algorithm for Online Matching on a Line

et al. 2019

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“…They gave two conjectures that the competitive ratio of this problem is 9 and that the Work-Function algorithm has a constant competitive ratio, both of which were later disproved in [11] and [5], respectively. This problem was studied in [2,3,6,[14][15][16], and the best known deterministic algorithm is the Robust Matching algorithm [15], which is Θ(log n)-competitive [14,16].…”

Section: Related Workmentioning

confidence: 99%

“…In this section, we show lower bounds on the competitive ratio of OFAL(k) for k = 3, 4, and 5. To simplify the proofs, we use Definitions 4 and 5 and Proposition 1, observed in [3,11], that allow us to restrict online algorithms to consider.…”

Section: Online Facility Assignment Problem On Linementioning

confidence: 99%

Competitive Analysis for Two Variants of Online Metric Matching Problem

Itoh

Miyazaki

Satake

2020

Combinatorial Optimization and Applications

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In this paper, we study two variants of the online metric matching problem. The first problem is the online metric matching problem where all the servers are placed at one of two positions in the metric space. We show that a simple greedy algorithm achieves the competitive ratio of 3 and give a matching lower bound. The second problem is the online facility assignment problem on a line, where servers have capacities, servers and requests are placed on 1-dimensional line, and the distances between any two consecutive servers are the same. We show lower bounds 1+ √ 6 (> 3.44948), 4+ √ 73 3 (> 4.18133) and 13 3 (> 4.33333) on the competitive ratio when the numbers of servers are 3, 4 and 5, respectively.

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“…Good algorithms had been elusive until recently, and the current best-known algorithm for OMM-LINE is a deterministic O(log k)-competitive algorithm [Rag18], prior to which the best-known results were an O(log k) upper bound for randomized algorithms [GL12] and an O(log 2 k) upper bound for deterministic algorithms [NR17]. There also exists an Ω(log k) lower bound on natural families of algorithms for OMM-LINE [AFT18,KN03] making this an intriguing open problem.…”

Section: Introductionmentioning

confidence: 99%

PERMUTATION Strikes Back: The Power of Recourse in Online Metric Matching

Gupta¹,

Krishnaswamy²,

Sandeep³

2019

Preprint

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In the classical ONLINE-METRIC-MATCHING problem, we are given a metric space with k servers. A collection of clients arrive in an online fashion, and upon arrival, a client should irrevocably be matched to an as-yet-unmatched server. The goal is to find an online matching which minimizes the total cost, i.e., the sum of distances between each client and the server it is matched to. We know deterministic algorithms [KP93, KMV94] that achieve a competitive ratio of 2k−1, and this bound is tight for deterministic algorithms. Randomization can be used to overcome this lower bound, and we know O(log 2 k) competitive algorithms [BBGN07] and an Ω(log k) lower bound. The problem has also long been considered in specialized metrics such as the line metric or metrics of bounded doubling dimension, with the current best result on a line metric being a deterministic O(log k) competitive algorithm [Rag18]. Obtaining (or refuting) O(log k)-competitive algorithms in general metrics and constant-competitive algorithms on the line metric have been long-standing open questions in this area.In this paper, we investigate the robustness of these lower bounds by considering the Online Metric Matching with Recourse problem where we are allowed to change a small number of previous assignments upon arrival of a new client. Indeed, we show that a small logarithmic amount of recourse can significantly improve the quality of matchings we can maintain. For general metrics, we show a simple deterministic O(log k)-competitive algorithm with O(log k)-amortized recourse, an exponential improvement over the 2k − 1 lower bound when no recourse is allowed. We next consider the line metric, and present a deterministic algorithm which is 3-competitive and has O(log k)-recourse, again a substantial improvement over the best known O(log k)-competitive algorithm when no recourse is allowed. We finally illustrate another benefit of allowing limited recourse: we can extend the Online Metric Matching model to handle arrivals and departures of both clients and servers (as opposed to just handling arrivals of clients) and still maintain competitive solutions. Indeed, we show a simple randomized O(log n)-competitive algorithm with O(log ∆)-recourse in this fully online setting, where n is the number of points in the metric space and ∆ is the aspect ratio of the underlying metric. Perhaps the most important technical contribution of this work is in showing that these improved results can, in fact, be achieved by suitably equipping a two-decades-old algorithm PERMUTATION for online metric matching with limited recourse.

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A Collection of Lower Bounds for Online Matching on the Line

Cited by 16 publications

References 17 publications

A $${o}\mathopen {}\left( n\right) \mathclose {}$$-Competitive Deterministic Algorithm for Online Matching on a Line

A $${o}\mathopen {}\left( n\right) \mathclose {}$$-Competitive Deterministic Algorithm for Online Matching on a Line

Competitive Analysis for Two Variants of Online Metric Matching Problem

PERMUTATION Strikes Back: The Power of Recourse in Online Metric Matching

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