Deterministic Min-Cost Matching with Delays

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 for points in space are submitted in different times in this space by an adversary. The goal is to match requests, while minim...

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Veröffentlicht in:Theory of computing systems Jg. 64; H. 4; S. 572 - 592
Hauptverfasser: Azar, Yossi, Jacob Fanani, Amit
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York Springer US 01.05.2020
Springer Nature B.V
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ISSN:1432-4350, 1433-0490
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Abstract 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 for points in space 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 1 𝜖 m log 2 3 2 + 𝜖 -competitive deterministic algorithms for both problems and for any fixed 𝜖 > 0. In particular, for a small enough 𝜖 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. We also show that the analysis of our algorithms is tight. Our algorithms do not need to know the metric space in advance.
AbstractList 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 for points in space 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 1 𝜖 m log 2 3 2 + 𝜖 -competitive deterministic algorithms for both problems and for any fixed 𝜖 > 0. In particular, for a small enough 𝜖 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. We also show that the analysis of our algorithms is tight. Our algorithms do not need to know the metric space in advance.
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 for points in space 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 Om-competitive algorithm, m being the number of requests. We introduce O1ðoe-mlog232+ðoe--competitive deterministic algorithms for both problems and for any fixed ðoe- > 0. In particular, for a small enough ðoe- the competitive ratio becomes Om0.59. These are the first deterministic algorithms for the mentioned online matching problems, achieving a sub-linear competitive ratio. We also show that the analysis of our algorithms is tight. Our algorithms do not need to know the metric space in advance.
Author Azar, Yossi
Jacob Fanani, Amit
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Keywords Matching
Delayed service
Bipartite matching
Online algorithm
Competitive analysis
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References_xml – reference: Azar, Y., Ganesh, A., Ge, R., Panigrahi, D.: Online service with delay. In: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pp. 551–563 (2017)
– reference: KalyanasundaramBPruhsKOnline weighted matchingJ. Algorithms1993143478488121312110.1006/jagm.1993.1026
– reference: Dantzig, G.B.: Linear programming and extensions. Princeton University Press (1963)
– reference: Khuller, S., Mitchell, S.G., Vazirani, V.V.: On-Line Algorithms for Weighted Bipartite Matching and Stable Marriages. In: Automata, Languages and Programming, 18Th International Colloquium, pp. 728–738 (1991)
– reference: Nayyar, K., Raghvendra, S.: An input sensitive online algorithm for the metric bipartite matching problem. In: 58Th IEEE Annual Symposium on Foundations of Computer Science, pp. 505–515 (2017)
– reference: Raghvendra, S.: A robust and optimal online algorithm for minimum metric bipartite matching. In: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pp. 18:1–18:16 (2016)
– reference: FuchsBHochstättlerWKernWOnline matching on a lineTheor. Comput. Sci.20053321-3251264212250510.1016/j.tcs.2004.10.028
– reference: Meyerson, A., Nanavati, A., Poplawski, L.J.: Randomized online algorithms for minimum metric bipartite matching. In: Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 954–959 (2006)
– reference: Emek, Y., Kutten, S., Wattenhofer, R.: Online matching: haste makes waste!. In: Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, pp. 333–344 (2016)
– reference: Emek, Y., Shapiro, Y., Wang, Y.: Minimum cost perfect matching with delays for Two sources. In: Algorithms and Complexity - 10Th International Conference, pp. 209–221 (2017)
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Snippet 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...
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SubjectTerms Algorithms
Competition
Computer Science
Matching
Metric space
Special Issue on Approximation and Online Algorithms 2018
Theory of Computation
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Title Deterministic Min-Cost Matching with Delays
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