Semi-streaming algorithms for submodular matroid intersection
While the basic greedy algorithm gives a semi-streaming algorithm with an approximation guarantee of 2 for the unweighted matching problem, it was only recently that Paz and Schwartzman obtained an analogous result for weighted instances. Their approach is based on the versatile local ratio techniqu...
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| Published in | Mathematical programming Vol. 197; no. 2; pp. 967 - 990 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.02.2023
Springer Springer Nature B.V |
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| Abstract | While the basic greedy algorithm gives a semi-streaming algorithm with an approximation guarantee of 2 for the
unweighted
matching problem, it was only recently that Paz and Schwartzman obtained an analogous result for weighted instances. Their approach is based on the versatile local ratio technique and also applies to generalizations such as weighted hypergraph matchings. However, the framework for the analysis fails for the related problem of weighted matroid intersection and as a result the approximation guarantee for weighted instances did not match the factor 2 achieved by the greedy algorithm for unweighted instances.Our main result closes this gap by developing a semi-streaming algorithm with an approximation guarantee of
2
+
ε
for
weighted
matroid intersection, improving upon the previous best guarantee of
4
+
ε
. Our techniques also allow us to generalize recent results by Levin and Wajc on submodular maximization subject to matching constraints to that of matroid-intersection constraints. While our algorithm is an adaptation of the local ratio technique used in previous works, the analysis deviates significantly and relies on structural properties of matroid intersection, called kernels. Finally, we also conjecture that our algorithm gives a
(
k
+
ε
)
approximation for the intersection of
k
matroids but prove that new tools are needed in the analysis as the structural properties we use fail for
k
≥
3
. |
|---|---|
| AbstractList | While the basic greedy algorithm gives a semi-streaming algorithm with an approximation guarantee of 2 for the unweighted matching problem, it was only recently that Paz and Schwartzman obtained an analogous result for weighted instances. Their approach is based on the versatile local ratio technique and also applies to generalizations such as weighted hypergraph matchings. However, the framework for the analysis fails for the related problem of weighted matroid intersection and as a result the approximation guarantee for weighted instances did not match the factor 2 achieved by the greedy algorithm for unweighted instances.Our main result closes this gap by developing a semi-streaming algorithm with an approximation guarantee of 2+ε for weighted matroid intersection, improving upon the previous best guarantee of 4+ε. Our techniques also allow us to generalize recent results by Levin and Wajc on submodular maximization subject to matching constraints to that of matroid-intersection constraints. While our algorithm is an adaptation of the local ratio technique used in previous works, the analysis deviates significantly and relies on structural properties of matroid intersection, called kernels. Finally, we also conjecture that our algorithm gives a (k+ε) approximation for the intersection of k matroids but prove that new tools are needed in the analysis as the structural properties we use fail for k≥3. While the basic greedy algorithm gives a semi-streaming algorithm with an approximation guarantee of 2 for the unweighted matching problem, it was only recently that Paz and Schwartzman obtained an analogous result for weighted instances. Their approach is based on the versatile local ratio technique and also applies to generalizations such as weighted hypergraph matchings. However, the framework for the analysis fails for the related problem of weighted matroid intersection and as a result the approximation guarantee for weighted instances did not match the factor 2 achieved by the greedy algorithm for unweighted instances.Our main result closes this gap by developing a semi-streaming algorithm with an approximation guarantee of $$2+\varepsilon $$ 2 + ε for weighted matroid intersection, improving upon the previous best guarantee of $$4+\varepsilon $$ 4 + ε . Our techniques also allow us to generalize recent results by Levin and Wajc on submodular maximization subject to matching constraints to that of matroid-intersection constraints. While our algorithm is an adaptation of the local ratio technique used in previous works, the analysis deviates significantly and relies on structural properties of matroid intersection, called kernels. Finally, we also conjecture that our algorithm gives a $$(k+\varepsilon )$$ ( k + ε ) approximation for the intersection of k matroids but prove that new tools are needed in the analysis as the structural properties we use fail for $$k\ge 3$$ k ≥ 3 . While the basic greedy algorithm gives a semi-streaming algorithm with an approximation guarantee of 2 for the matching problem, it was only recently that Paz and Schwartzman obtained an analogous result for weighted instances. Their approach is based on the versatile local ratio technique and also applies to generalizations such as weighted hypergraph matchings. However, the framework for the analysis fails for the related problem of weighted matroid intersection and as a result the approximation guarantee for weighted instances did not match the factor 2 achieved by the greedy algorithm for unweighted instances.Our main result closes this gap by developing a semi-streaming algorithm with an approximation guarantee of for matroid intersection, improving upon the previous best guarantee of . Our techniques also allow us to generalize recent results by Levin and Wajc on submodular maximization subject to matching constraints to that of matroid-intersection constraints. While our algorithm is an adaptation of the local ratio technique used in previous works, the analysis deviates significantly and relies on structural properties of matroid intersection, called kernels. Finally, we also conjecture that our algorithm gives a approximation for the intersection of matroids but prove that new tools are needed in the analysis as the structural properties we use fail for . While the basic greedy algorithm gives a semi-streaming algorithm with an approximation guarantee of 2 for the unweighted matching problem, it was only recently that Paz and Schwartzman obtained an analogous result for weighted instances. Their approach is based on the versatile local ratio technique and also applies to generalizations such as weighted hypergraph matchings. However, the framework for the analysis fails for the related problem of weighted matroid intersection and as a result the approximation guarantee for weighted instances did not match the factor 2 achieved by the greedy algorithm for unweighted instances.Our main result closes this gap by developing a semi-streaming algorithm with an approximation guarantee of 2 + ε for weighted matroid intersection, improving upon the previous best guarantee of 4 + ε . Our techniques also allow us to generalize recent results by Levin and Wajc on submodular maximization subject to matching constraints to that of matroid-intersection constraints. While our algorithm is an adaptation of the local ratio technique used in previous works, the analysis deviates significantly and relies on structural properties of matroid intersection, called kernels. Finally, we also conjecture that our algorithm gives a ( k + ε ) approximation for the intersection of k matroids but prove that new tools are needed in the analysis as the structural properties we use fail for k ≥ 3 . While the basic greedy algorithm gives a semi-streaming algorithm with an approximation guarantee of 2 for the unweighted matching problem, it was only recently that Paz and Schwartzman obtained an analogous result for weighted instances. Their approach is based on the versatile local ratio technique and also applies to generalizations such as weighted hypergraph matchings. However, the framework for the analysis fails for the related problem of weighted matroid intersection and as a result the approximation guarantee for weighted instances did not match the factor 2 achieved by the greedy algorithm for unweighted instances.Our main result closes this gap by developing a semi-streaming algorithm with an approximation guarantee of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2+\varepsilon $$\end{document} 2 + ε for weighted matroid intersection, improving upon the previous best guarantee of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4+\varepsilon $$\end{document} 4 + ε . Our techniques also allow us to generalize recent results by Levin and Wajc on submodular maximization subject to matching constraints to that of matroid-intersection constraints. While our algorithm is an adaptation of the local ratio technique used in previous works, the analysis deviates significantly and relies on structural properties of matroid intersection, called kernels. Finally, we also conjecture that our algorithm gives a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(k+\varepsilon )$$\end{document} ( k + ε ) approximation for the intersection of k matroids but prove that new tools are needed in the analysis as the structural properties we use fail for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k\ge 3$$\end{document} k ≥ 3 . While the basic greedy algorithm gives a semi-streaming algorithm with an approximation guarantee of 2 for the unweighted matching problem, it was only recently that Paz and Schwartzman obtained an analogous result for weighted instances. Their approach is based on the versatile local ratio technique and also applies to generalizations such as weighted hypergraph matchings. However, the framework for the analysis fails for the related problem of weighted matroid intersection and as a result the approximation guarantee for weighted instances did not match the factor 2 achieved by the greedy algorithm for unweighted instances.Our main result closes this gap by developing a semi-streaming algorithm with an approximation guarantee of 2 + ε for weighted matroid intersection, improving upon the previous best guarantee of 4 + ε . Our techniques also allow us to generalize recent results by Levin and Wajc on submodular maximization subject to matching constraints to that of matroid-intersection constraints. While our algorithm is an adaptation of the local ratio technique used in previous works, the analysis deviates significantly and relies on structural properties of matroid intersection, called kernels. Finally, we also conjecture that our algorithm gives a ( k + ε ) approximation for the intersection of k matroids but prove that new tools are needed in the analysis as the structural properties we use fail for k ≥ 3 .While the basic greedy algorithm gives a semi-streaming algorithm with an approximation guarantee of 2 for the unweighted matching problem, it was only recently that Paz and Schwartzman obtained an analogous result for weighted instances. Their approach is based on the versatile local ratio technique and also applies to generalizations such as weighted hypergraph matchings. However, the framework for the analysis fails for the related problem of weighted matroid intersection and as a result the approximation guarantee for weighted instances did not match the factor 2 achieved by the greedy algorithm for unweighted instances.Our main result closes this gap by developing a semi-streaming algorithm with an approximation guarantee of 2 + ε for weighted matroid intersection, improving upon the previous best guarantee of 4 + ε . Our techniques also allow us to generalize recent results by Levin and Wajc on submodular maximization subject to matching constraints to that of matroid-intersection constraints. While our algorithm is an adaptation of the local ratio technique used in previous works, the analysis deviates significantly and relies on structural properties of matroid intersection, called kernels. Finally, we also conjecture that our algorithm gives a ( k + ε ) approximation for the intersection of k matroids but prove that new tools are needed in the analysis as the structural properties we use fail for k ≥ 3 . While the basic greedy algorithm gives a semi-streaming algorithm with an approximation guarantee of 2 for the unweighted matching problem, it was only recently that Paz and Schwartzman obtained an analogous result for weighted instances. Their approach is based on the versatile local ratio technique and also applies to generalizations such as weighted hypergraph matchings. However, the framework for the analysis fails for the related problem of weighted matroid intersection and as a result the approximation guarantee for weighted instances did not match the factor 2 achieved by the greedy algorithm for unweighted instances.Our main result closes this gap by developing a semi-streaming algorithm with an approximation guarantee of [Formula omitted] for weighted matroid intersection, improving upon the previous best guarantee of [Formula omitted]. Our techniques also allow us to generalize recent results by Levin and Wajc on submodular maximization subject to matching constraints to that of matroid-intersection constraints. While our algorithm is an adaptation of the local ratio technique used in previous works, the analysis deviates significantly and relies on structural properties of matroid intersection, called kernels. Finally, we also conjecture that our algorithm gives a [Formula omitted] approximation for the intersection of k matroids but prove that new tools are needed in the analysis as the structural properties we use fail for [Formula omitted]. |
| Audience | Academic |
| Author | Jordan, Linus Svensson, Ola Garg, Paritosh |
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| Keywords | 68W27 05B35 68W25 Matroid Intersection Submodular Functions Semi-Streaming Algorithms |
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| References | MuthukrishnanSData streams: algorithms and applicationsFound. Trends Theor. Comput. Sci.200512117236237950710.1561/04000000021128.68025 PazASchwartzmanGA (2+ ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document})-approximation for maximum weight matching in the semi-streaming modelACM Transactions on Algorithms (TALG)20181521153951091 SchrijverACombinatorial Optimization: Polyhedra and Efficiency2003BerlinSpringer1041.90001 Buchbinder, N., Feldman, M., Naor, J., Schwartz, R.: Submodular maximization with cardinality constraints, In: Proceedings of the twenty-fifth annual ACM-SIAM symposium on Discrete algorithms, SIAM, pp. 1433–1452 (2014) Ghaffari, M., Wajc, D.: Simplified and Space-Optimal Semi-Streaming (2+epsilon)-Approximate Matching, 2nd Symposium on Simplicity in Algorithms (SOSA 2019) (Dagstuhl, Germany) (Jeremy T. Fineman and Michael Mitzenmacher, eds.), OpenAccess Series in Informatics (OASIcs), vol. 69, Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, pp. 13:1–13:8 (2018) Huang, C.-C., Kakimura, N., Mauras, S., Yoshida, Y.: Approximability of monotone submodular function maximization under cardinality and matroid constraints in the streaming model, CoRR arXiv:2002.05477 (2020) Levin, R., Wajc, D.: Streaming submodular matching meets the primal-dual method, In: Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM, pp. 1914–1933 (2021) ChakrabartiAKaleSSubmodular maximization meets streaming: Matchings, matroids, and moreMath. Program.20151541225247342193410.1007/s10107-015-0900-71342.90212 EdmondsJMatroid intersection, Discrete Optimization I1979Amsterdam, NetherlandsElsevier39490416.05025 McGregor, A.: Finding graph matchings in data streams, Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques (Berlin, Heidelberg) (Chandra Chekuri, Klaus Jansen, José D. P. Rolim, and Luca Trevisan, eds.), Springer Berlin Heidelberg, pp. 170–181 (2005) FeigenbaumJKannanSMcGregorASuriSZhangJOn graph problems in a semi-streaming modelTheoret. Comput. Sci.20053482–3207216218137610.1016/j.tcs.2005.09.0131081.68069 Bar-YehudaRBendelKFreundARawitzDLocal ratio: A unified framework for approximation algorithms. in memoriam: Shimon even 1935-2004ACM Computing Surveys (CSUR)20043642246310.1145/1041680.1041683 FleinerTA Matroid Generalization of the Stable Matching Polytope2001BerlinSpringer1051141010.90062 Bar-Yehuda, R., Even, S.: A local-ratio theorem for approximating the weighted vertex cover problem, Analysis and Design of Algorithms for Combinatorial Problems (G. Ausiello and M. Lucertini, eds.), North-Holland Mathematics Studies, vol. 109, North-Holland, pp. 27 – 45 (1985) CrouchMStubbsDMImproved streaming algorithms for weighted matching, via unweighted matchingLeibniz Int. Proc. Inform. LIPIcs2014289610433189841359.68306 1858_CR1 1858_CR3 T Fleiner (1858_CR8) 2001 J Feigenbaum (1858_CR7) 2005; 348 A Schrijver (1858_CR15) 2003 1858_CR12 1858_CR11 1858_CR10 J Edmonds (1858_CR6) 1979 M Crouch (1858_CR5) 2014; 28 A Chakrabarti (1858_CR4) 2015; 154 S Muthukrishnan (1858_CR13) 2005; 1 A Paz (1858_CR14) 2018; 15 1858_CR9 R Bar-Yehuda (1858_CR2) 2004; 36 |
| References_xml | – reference: MuthukrishnanSData streams: algorithms and applicationsFound. Trends Theor. Comput. Sci.200512117236237950710.1561/04000000021128.68025 – reference: Ghaffari, M., Wajc, D.: Simplified and Space-Optimal Semi-Streaming (2+epsilon)-Approximate Matching, 2nd Symposium on Simplicity in Algorithms (SOSA 2019) (Dagstuhl, Germany) (Jeremy T. Fineman and Michael Mitzenmacher, eds.), OpenAccess Series in Informatics (OASIcs), vol. 69, Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, pp. 13:1–13:8 (2018) – reference: Levin, R., Wajc, D.: Streaming submodular matching meets the primal-dual method, In: Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM, pp. 1914–1933 (2021) – reference: McGregor, A.: Finding graph matchings in data streams, Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques (Berlin, Heidelberg) (Chandra Chekuri, Klaus Jansen, José D. P. Rolim, and Luca Trevisan, eds.), Springer Berlin Heidelberg, pp. 170–181 (2005) – reference: Buchbinder, N., Feldman, M., Naor, J., Schwartz, R.: Submodular maximization with cardinality constraints, In: Proceedings of the twenty-fifth annual ACM-SIAM symposium on Discrete algorithms, SIAM, pp. 1433–1452 (2014) – reference: ChakrabartiAKaleSSubmodular maximization meets streaming: Matchings, matroids, and moreMath. Program.20151541225247342193410.1007/s10107-015-0900-71342.90212 – reference: PazASchwartzmanGA (2+ ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document})-approximation for maximum weight matching in the semi-streaming modelACM Transactions on Algorithms (TALG)20181521153951091 – reference: EdmondsJMatroid intersection, Discrete Optimization I1979Amsterdam, NetherlandsElsevier39490416.05025 – reference: FeigenbaumJKannanSMcGregorASuriSZhangJOn graph problems in a semi-streaming modelTheoret. Comput. Sci.20053482–3207216218137610.1016/j.tcs.2005.09.0131081.68069 – reference: Bar-YehudaRBendelKFreundARawitzDLocal ratio: A unified framework for approximation algorithms. in memoriam: Shimon even 1935-2004ACM Computing Surveys (CSUR)20043642246310.1145/1041680.1041683 – reference: SchrijverACombinatorial Optimization: Polyhedra and Efficiency2003BerlinSpringer1041.90001 – reference: FleinerTA Matroid Generalization of the Stable Matching Polytope2001BerlinSpringer1051141010.90062 – reference: Bar-Yehuda, R., Even, S.: A local-ratio theorem for approximating the weighted vertex cover problem, Analysis and Design of Algorithms for Combinatorial Problems (G. Ausiello and M. Lucertini, eds.), North-Holland Mathematics Studies, vol. 109, North-Holland, pp. 27 – 45 (1985) – reference: CrouchMStubbsDMImproved streaming algorithms for weighted matching, via unweighted matchingLeibniz Int. Proc. Inform. LIPIcs2014289610433189841359.68306 – reference: Huang, C.-C., Kakimura, N., Mauras, S., Yoshida, Y.: Approximability of monotone submodular function maximization under cardinality and matroid constraints in the streaming model, CoRR arXiv:2002.05477 (2020) – volume-title: Combinatorial Optimization: Polyhedra and Efficiency year: 2003 ident: 1858_CR15 – start-page: 105 volume-title: A Matroid Generalization of the Stable Matching Polytope year: 2001 ident: 1858_CR8 doi: 10.1007/3-540-45535-3_9 – ident: 1858_CR1 doi: 10.1137/1.9781611973730.80 – ident: 1858_CR9 – ident: 1858_CR11 doi: 10.1137/1.9781611976465.114 – volume: 28 start-page: 96 year: 2014 ident: 1858_CR5 publication-title: Leibniz Int. Proc. Inform. LIPIcs – volume: 15 start-page: 1 issue: 2 year: 2018 ident: 1858_CR14 publication-title: ACM Transactions on Algorithms (TALG) – ident: 1858_CR12 doi: 10.1007/11538462_15 – ident: 1858_CR10 – volume: 154 start-page: 225 issue: 1 year: 2015 ident: 1858_CR4 publication-title: Math. Program. doi: 10.1007/s10107-015-0900-7 – volume: 348 start-page: 207 issue: 2–3 year: 2005 ident: 1858_CR7 publication-title: Theoret. Comput. Sci. doi: 10.1016/j.tcs.2005.09.013 – volume: 1 start-page: 117 issue: 2 year: 2005 ident: 1858_CR13 publication-title: Found. Trends Theor. Comput. Sci. doi: 10.1561/0400000002 – ident: 1858_CR3 doi: 10.1016/S0304-0208(08)73101-3 – start-page: 39 volume-title: Matroid intersection, Discrete Optimization I year: 1979 ident: 1858_CR6 – volume: 36 start-page: 422 year: 2004 ident: 1858_CR2 publication-title: ACM Computing Surveys (CSUR) doi: 10.1145/1041680.1041683 |
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| Snippet | While the basic greedy algorithm gives a semi-streaming algorithm with an approximation guarantee of 2 for the
unweighted
matching problem, it was only... While the basic greedy algorithm gives a semi-streaming algorithm with an approximation guarantee of 2 for the matching problem, it was only recently that Paz... While the basic greedy algorithm gives a semi-streaming algorithm with an approximation guarantee of 2 for the unweighted matching problem, it was only... |
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| SubjectTerms | Algorithms Analysis Approximation Calculus of Variations and Optimal Control; Optimization Combinatorics Full Length Paper Greedy algorithms Matching Mathematical analysis Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Theoretical |
| Title | Semi-streaming algorithms for submodular matroid intersection |
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