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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

In the matroid partitioning problem, we are given k matroids ℳ₁ = (V, ℐ_1), … , ℳ_k = (V, ℐ_k) defined over a common ground set V of n elements, and we need to find a partitionable set S ⊆ V of largest possible cardinality, denoted by p. Here, a set S ⊆ V is called partitionable if there exists a partition (S_1, … , S_k) of S with S_i ∈ ℐ_i for i = 1, …, k. In 1986, Cunningham [Cunningham, 1986] presented a matroid partition algorithm that uses O(n p^{3/2} + k n) independence oracle queries, which was the previously known best algorithm. This query complexity is O(n^{5/2}) when k ≤ n.
Our main result is to present a matroid partition algorithm that uses Õ(k^{1/3} n p + k n) independence oracle queries, which is Õ(n^{7/3}) when k ≤ n. This improves upon previous Cunningham’s algorithm. To obtain this, we present a new approach edge recycling augmentation, which can be attained through new ideas: an efficient utilization of the binary search technique by Nguyễn [Nguyen, 2019] and Chakrabarty-Lee-Sidford-Singla-Wong [Chakrabarty et al., 2019] and a careful analysis of the number of independence oracle queries. Our analysis differs significantly from the one for matroid intersection algorithms, because of the parameter k. We also present a matroid partition algorithm that uses Õ((n + k) √p) rank oracle queries.

Tatsuya Terao. Faster Matroid Partition Algorithms. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 104:1-104:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{terao:LIPIcs.ICALP.2023.104, author = {Terao, Tatsuya}, title = {{Faster Matroid Partition Algorithms}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {104:1--104:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.104}, URN = {urn:nbn:de:0030-drops-181566}, doi = {10.4230/LIPIcs.ICALP.2023.104}, annote = {Keywords: Matroid Partition, Matroid Union, Combinatorial Optimization} }

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**Published in:** LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022)

For an undirected graph G and distinct vertices s₁, t₁, … , s_k, t_k called terminals, the shortest k-disjoint paths problem asks for k pairwise vertex-disjoint paths P₁, … , P_k such that P_i connects s_i and t_i for i = 1, … , k and the sum of their lengths is minimized. This problem is a natural optimization version of the well-known k-disjoint paths problem, and its polynomial solvability is widely open. One of the best results on the shortest k-disjoint paths problem is due to Datta et al. [Datta et al., 2018], who present a polynomial-time algorithm for the case when G is planar and all the terminals are on one face. In this paper, we extend this result by giving a polynomial-time randomized algorithm for the case when all the terminals except one are on some face of G. In our algorithm, we combine the arguments of Datta et al. with some results on the shortest disjoint (A + B)-paths problem shown by Hirai and Namba [Hirai and Namba, 2018]. To this end, we present a non-trivial bijection between k disjoint paths and disjoint (A + B)-paths, which is a key technical contribution of this paper.

Yusuke Kobayashi and Tatsuya Terao. One-Face Shortest Disjoint Paths with a Deviation Terminal. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 47:1-47:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{kobayashi_et_al:LIPIcs.ISAAC.2022.47, author = {Kobayashi, Yusuke and Terao, Tatsuya}, title = {{One-Face Shortest Disjoint Paths with a Deviation Terminal}}, booktitle = {33rd International Symposium on Algorithms and Computation (ISAAC 2022)}, pages = {47:1--47:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-258-7}, ISSN = {1868-8969}, year = {2022}, volume = {248}, editor = {Bae, Sang Won and Park, Heejin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.47}, URN = {urn:nbn:de:0030-drops-173322}, doi = {10.4230/LIPIcs.ISAAC.2022.47}, annote = {Keywords: shortest disjoint paths, polynomial time algorithm, planar graph} }

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