Blikstad, J., Mukhopadhyay, S. orcid.org/0000-0002-3722-4679, Nanongkai, D. et al. (1 more author) (2023) Fast algorithms via dynamic-oracle matroids. In: Barna, S. and Servedio, R.A., (eds.) STOC 2023: Proceedings of the 55th Annual ACM Symposium on Theory of Computing. STOC 2023: 55th Annual ACM Symposium on Theory of Computing, 20-23 Jun 2023, Orlando, Florida. ACM , pp. 1229-1242. ISBN 9781450399135
Abstract
We initiate the study of matroid problems in a new oracle model called dynamic oracle. Our algorithms in this model lead to new bounds for some classic problems, and a "unified"algorithm whose performance matches previous results developed in various papers for various problems. We also show a lower bound that answers some open problems from a few decades ago. Concretely, our results are as follows. Improved algorithms for matroid union and disjoint spanning trees. We show an algorithm with Õk(n+rr) dynamic-rank-query and time complexities for the matroid union problem over k matroids, where n is the input size, r is the output size, and Õk hides poly(k, log(n)). This implies the following consequences. (i) An improvement over the Õk(nr) bound implied by [Chakrabarty-Lee-Sidford-Singla-Wong FOCS'19] for matroid union in the traditional rank-query model. (ii) An Õk(|E|+|V||V|)-time algorithm for the k-disjoint spanning tree problem. This is nearly linear for moderately dense input graphs and improves the Õk(|V||E|) bounds of Gabow-Westermann [STOC'88] and Gabow [STOC'91]. Consequently, this gives improved bounds for, e.g., Shannon Switching Game and Graph Irreducibility. Matroid intersection. We show a matroid intersection algorithm with Õ(nr) dynamic-rank-query and time complexities. This implies new bounds for some problems (e.g. maximum forest with deadlines) and bounds that match the classic ones obtained in various papers for various problems, e.g. colorful spanning tree [Gabow-Stallmann ICALP'85], graphic matroid intersection [Gabow-Xu FOCS'89], simple job scheduling matroid intersection [Xu-Gabow ISAAC'94], and Hopcroft-Karp combinatorial bipartite matching. More importantly, this is done via a "unified"algorithm in the sense that an improvement over our dynamic-rank-query algorithm would imply improved bounds for all the above problems simultaneously. Lower bounds. We show simple super-linear (ω(nlogn)) query lower bounds for matroid intersection and union problems in our dynamic-rank-oracle and the traditional independence-query models; the latter improves the previous log2(3)n-o(n) bound by Harvey [SODA'08] and answers an open problem raised by, e.g., Welsh [1976] and CLSSW [FOCS'19].
Metadata
Item Type: | Proceedings Paper |
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Authors/Creators: |
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Editors: |
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Copyright, Publisher and Additional Information: | © 2023 ACM. Publication rights licensed to ACM. ACM acknowledges that this contribution was authored or co-authored by an employee, contractor or affiliate of a national government. As such, the Government retains a nonexclusive, royalty-free right to publish or reproduce this article, or to allow others to do so, for Government purposes only. |
Keywords: | matroid union; combinatorial optimization; arboricity; spanning tree packing; matroids; matroid intersection; dynamic algorithms |
Dates: |
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Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Engineering (Sheffield) > Department of Computer Science (Sheffield) |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 01 Dec 2023 15:41 |
Last Modified: | 01 Dec 2023 15:41 |
Published Version: | http://dx.doi.org/10.1145/3564246.3585219 |
Status: | Published |
Publisher: | ACM |
Refereed: | Yes |
Identification Number: | 10.1145/3564246.3585219 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:205822 |