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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| 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 O1ðoe-mlog232+ðoe--competitive deterministic algorithms for both problems and for any fixed ðoe- > 0. In particular, for a small enough ðoe- 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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| References | BansalNBuchbinderNGuptaANaorJA randomized O(log2k)-competitive algorithm for metric bipartite matchingAlgorithmica2014682390403315205410.1007/s00453-012-9676-9 Bienkowski, M., Kraska, A., Schmidt, P.: A match in time saves nine: deterministic online matching with delays. In: Approximation and Online Algorithms - 15Th International Workshop, pp. 132–146 (2017) Dantzig, G.B.: Linear programming and extensions. Princeton University Press (1963) FuchsBHochstättlerWKernWOnline matching on a lineTheor. Comput. Sci.20053321-3251264212250510.1016/j.tcs.2004.10.028 Gupta, A., Lewi, K.: The online metric matching problem for doubling metrics. In: Automata, Languages, and Programming - 39Th International Colloquium, pp. 424–435 (2012) 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) Antoniadis, A., Barcelo, N., Nugent, M., Pruhs, K., Scquizzato, M.: A O(N)-competitive deterministic algorithm for online matching on a line. In: Approximation and Online Algorithms - 12Th International Workshop, pp. 11–22 (2014) 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) 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) Koutsoupias, E., Nanavati, A.: The online matching problem on a line. In: Approximation and Online Algorithms, First International Workshop, pp. 179–191 (2003) 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) 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) ReingoldEMTarjanREOn a greedy heuristic for complete matchingSIAM J. Comput.198110467668163542510.1137/0210050 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) 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) KalyanasundaramBPruhsKOnline weighted matchingJ. Algorithms1993143478488121312110.1006/jagm.1993.1026 Ashlagi, I., Azar, Y., Charikar, M., Chiplunkar, A., Geri, O., Kaplan, H., Makhijani, R.M., Wang, Y., Wattenhofer, R.: Min-cost bipartite perfect matching with delays. In: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pp. 1:1–1:20 (2017) Azar, Y., Chiplunkar, A., Kaplan, H.: Polylogarithmic bounds on the competitiveness of min-cost perfect matching with delays. In: Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1051–1061 (2017) EdmondsJPaths, trees, and flowersCan. J. Math.19651744946717790710.4153/CJM-1965-045-4 Kalyanasundaram, B., Pruhs, K.: On-Line Network Optimization Problems. In: Online Algorithms, the State of the Art (The Book Grow out of a Dagstuhl Seminar), pp. 268–280 (1996) Bienkowski, M., Kraska, A., Liu, H.H., Schmidt, P.: A primal-dual online deterministic algorithm for matching with delays. arXiv:abs/1804.08097 (2018) 9963_CR7 J Edmonds (9963_CR9) 1965; 17 9963_CR11 9963_CR8 9963_CR10 9963_CR13 9963_CR6 9963_CR3 9963_CR15 9963_CR4 9963_CR1 9963_CR17 9963_CR2 9963_CR16 9963_CR19 EM Reingold (9963_CR21) 1981; 10 9963_CR18 N Bansal (9963_CR5) 2014; 68 B Kalyanasundaram (9963_CR14) 1993; 14 B Fuchs (9963_CR12) 2005; 332 9963_CR20 |
| 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) – reference: Gupta, A., Lewi, K.: The online metric matching problem for doubling metrics. In: Automata, Languages, and Programming - 39Th International Colloquium, pp. 424–435 (2012) – reference: Ashlagi, I., Azar, Y., Charikar, M., Chiplunkar, A., Geri, O., Kaplan, H., Makhijani, R.M., Wang, Y., Wattenhofer, R.: Min-cost bipartite perfect matching with delays. In: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pp. 1:1–1:20 (2017) – reference: Kalyanasundaram, B., Pruhs, K.: On-Line Network Optimization Problems. In: Online Algorithms, the State of the Art (The Book Grow out of a Dagstuhl Seminar), pp. 268–280 (1996) – reference: Bienkowski, M., Kraska, A., Schmidt, P.: A match in time saves nine: deterministic online matching with delays. In: Approximation and Online Algorithms - 15Th International Workshop, pp. 132–146 (2017) – reference: EdmondsJPaths, trees, and flowersCan. J. Math.19651744946717790710.4153/CJM-1965-045-4 – reference: BansalNBuchbinderNGuptaANaorJA randomized O(log2k)-competitive algorithm for metric bipartite matchingAlgorithmica2014682390403315205410.1007/s00453-012-9676-9 – reference: Bienkowski, M., Kraska, A., Liu, H.H., Schmidt, P.: A primal-dual online deterministic algorithm for matching with delays. arXiv:abs/1804.08097 (2018) – reference: Antoniadis, A., Barcelo, N., Nugent, M., Pruhs, K., Scquizzato, M.: A O(N)-competitive deterministic algorithm for online matching on a line. In: Approximation and Online Algorithms - 12Th International Workshop, pp. 11–22 (2014) – reference: ReingoldEMTarjanREOn a greedy heuristic for complete matchingSIAM J. Comput.198110467668163542510.1137/0210050 – reference: Koutsoupias, E., Nanavati, A.: The online matching problem on a line. In: Approximation and Online Algorithms, First International Workshop, pp. 179–191 (2003) – reference: Azar, Y., Chiplunkar, A., Kaplan, H.: Polylogarithmic bounds on the competitiveness of min-cost perfect matching with delays. In: Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1051–1061 (2017) – volume: 14 start-page: 478 issue: 3 year: 1993 ident: 9963_CR14 publication-title: J. Algorithms doi: 10.1006/jagm.1993.1026 – volume: 17 start-page: 449 year: 1965 ident: 9963_CR9 publication-title: Can. J. Math. doi: 10.4153/CJM-1965-045-4 – ident: 9963_CR8 doi: 10.7249/R366 – ident: 9963_CR15 doi: 10.1007/BFb0029573 – ident: 9963_CR7 doi: 10.1007/978-3-319-89441-6_11 – ident: 9963_CR1 doi: 10.1007/978-3-319-18263-6_2 – volume: 68 start-page: 390 issue: 2 year: 2014 ident: 9963_CR5 publication-title: Algorithmica doi: 10.1007/s00453-012-9676-9 – ident: 9963_CR6 doi: 10.1007/978-3-030-04693-4_4 – ident: 9963_CR11 doi: 10.1007/978-3-319-57586-5_18 – ident: 9963_CR3 doi: 10.1137/1.9781611974782.67 – ident: 9963_CR20 – ident: 9963_CR19 doi: 10.1109/FOCS.2017.53 – volume: 10 start-page: 676 issue: 4 year: 1981 ident: 9963_CR21 publication-title: SIAM J. Comput. doi: 10.1137/0210050 – ident: 9963_CR2 – ident: 9963_CR17 doi: 10.1007/978-3-540-24592-6_14 – ident: 9963_CR18 doi: 10.1145/1109557.1109662 – ident: 9963_CR4 doi: 10.1145/3055399.3055475 – ident: 9963_CR16 doi: 10.1007/3-540-54233-7_178 – ident: 9963_CR10 doi: 10.1145/2897518.2897557 – volume: 332 start-page: 251 issue: 1-3 year: 2005 ident: 9963_CR12 publication-title: Theor. Comput. Sci. doi: 10.1016/j.tcs.2004.10.028 – ident: 9963_CR13 doi: 10.1007/978-3-642-31594-7_36 |
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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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