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**Published in:** LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

The Subset Feedback Vertex Set problem (SFVS) is to delete k vertices from a given graph such that in the remaining graph, any vertex in a subset T of vertices (called a terminal set) is not in a cycle. The famous Feedback Vertex Set problem is the special case of SFVS with T being the whole set of vertices. In this paper, we study exact algorithms for SFVS in Split Graphs (SFVS-S) and SFVS in Chordal Graphs (SFVS-C). SFVS-S generalizes the minimum vertex cover problem and the prize-collecting version of the maximum independent set problem in hypergraphs (PCMIS), and SFVS-C further generalizes SFVS-S. Both SFVS-S and SFVS-C are implicit 3-Hitting Set problems. However, it is not easy to solve them faster than 3-Hitting Set. In 2019, Philip, Rajan, Saurabh, and Tale (Algorithmica 2019) proved that SFVS-C can be solved in 𝒪^*(2^k) time, slightly improving the best result 𝒪^*(2.0755^k) for 3-Hitting Set. In this paper, we break the "2^k-barrier" for SFVS-S and SFVS-C by introducing an 𝒪^*(1.8192^k)-time algorithm. This achievement also indicates that PCMIS can be solved in 𝒪^*(1.8192ⁿ) time, marking the first exact algorithm for PCMIS that outperforms the trivial 𝒪^*(2ⁿ) threshold. Our algorithm uses reduction and branching rules based on the Dulmage-Mendelsohn decomposition and a divide-and-conquer method.

Tian Bai and Mingyu Xiao. Breaking the Barrier 2^k for Subset Feedback Vertex Set in Chordal Graphs. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 15:1-15:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bai_et_al:LIPIcs.MFCS.2024.15, author = {Bai, Tian and Xiao, Mingyu}, title = {{Breaking the Barrier 2^k for Subset Feedback Vertex Set in Chordal Graphs}}, booktitle = {49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)}, pages = {15:1--15:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-335-5}, ISSN = {1868-8969}, year = {2024}, volume = {306}, editor = {Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.15}, URN = {urn:nbn:de:0030-drops-205711}, doi = {10.4230/LIPIcs.MFCS.2024.15}, annote = {Keywords: Subset Feedback Vertex Set, Prize-Collecting Maximum Independent Set, Parameterized Algorithms, Split Graphs, Chordal Graphs, Dulmage-Mendelsohn Decomposition} }

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**Published in:** LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

The paper introduces two player connectivity games played on finite bipartite graphs. Algorithms that solve these connectivity games can be used as subroutines for solving Müller games. Müller games constitute a well established class of games in model checking and verification. In connectivity games, the objective of one of the players is to visit every node of the game graph infinitely often. The first contribution of this paper is our proof that solving connectivity games can be reduced to the incremental strongly connected component maintenance (ISCCM) problem, an important problem in graph algorithms and data structures. The second contribution is that we non-trivially adapt two known algorithms for the ISCCM problem to provide two efficient algorithms that solve the connectivity games problem. Finally, based on the techniques developed, we recast Horn’s polynomial time algorithm that solves explicitly given Müller games and provide the first correctness proof of the algorithm. Our algorithms are more efficient than that of Horn’s algorithm. Our solution for connectivity games is used as a subroutine in the algorithm.

Zihui Liang, Bakh Khoussainov, Toru Takisaka, and Mingyu Xiao. Connectivity in the Presence of an Opponent. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 79:1-79:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{liang_et_al:LIPIcs.ESA.2023.79, author = {Liang, Zihui and Khoussainov, Bakh and Takisaka, Toru and Xiao, Mingyu}, title = {{Connectivity in the Presence of an Opponent}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {79:1--79:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.79}, URN = {urn:nbn:de:0030-drops-187324}, doi = {10.4230/LIPIcs.ESA.2023.79}, annote = {Keywords: Explicit M\"{u}ller games, games played on finite graphs, winning strategies, synthesis and analysis of games} }

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**Published in:** LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

The Traveling Tournament Problem (TTP) is a well-known benchmark problem in the field of tournament timetabling, which asks us to design a double round-robin schedule such that each pair of teams plays one game in each other’s home venue, minimizing the total distance traveled by all n teams (n is even). TTP-k is the problem with one more constraint that each team can have at most k consecutive home games or away games. The case where k = 3, TTP-3, is one of the most investigated cases. In this paper, we improve the approximation ratio of TTP-3 from (1.667+ε) to (1.598+ε), for any ε > 0. Previous schedules were constructed based on a Hamiltonian cycle of the graph. We propose a novel construction based on triangle packing. Then, by combining our triangle packing schedule with the Hamiltonian cycle schedule, we obtain the improved approximation ratio. The idea of our construction can also be extended to k ≥ 4. We demonstrate that the approximation ratio of TTP-4 can be improved from (1.750+ε) to (1.700+ε) by the same method. As an additional product, we also improve the approximation ratio of LDTTP-3 (TTP-3 where all teams are allocated on a straight line) from 4/3 to (6/5+ε).

Jingyang Zhao, Mingyu Xiao, and Chao Xu. Improved Approximation Algorithms for the Traveling Tournament Problem. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 83:1-83:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{zhao_et_al:LIPIcs.MFCS.2022.83, author = {Zhao, Jingyang and Xiao, Mingyu and Xu, Chao}, title = {{Improved Approximation Algorithms for the Traveling Tournament Problem}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {83:1--83:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-256-3}, ISSN = {1868-8969}, year = {2022}, volume = {241}, editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.83}, URN = {urn:nbn:de:0030-drops-168813}, doi = {10.4230/LIPIcs.MFCS.2022.83}, annote = {Keywords: Sports scheduling, Traveling Tournament Problem, Approximation algorithm} }

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Brief Announcement

**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

In the bounded-degree cut problem, we are given a multigraph G=(V,E), two disjoint vertex subsets A,B subseteq V, two functions u_A, u_B:V -> {0,1,...,|E|} on V, and an integer k >= 0. The task is to determine whether there is a minimal (A,B)-cut (V_A,V_B) of size at most k such that the degree of each vertex v in V_A in the induced subgraph G[V_A] is at most u_A(v) and the degree of each vertex v in V_B in the induced subgraph G[V_B] is at most u_B(v). In this paper, we show that the bounded-degree cut problem is fixed-parameter tractable by giving a 2^{18k}|G|^{O(1)}-time algorithm. This is the first single exponential FPT algorithm for this problem. The core of the algorithm lies two new lemmas based on important cuts, which give some upper bounds on the number of candidates for vertex subsets in one part of a minimal cut satisfying some properties. These lemmas can be used to design fixed-parameter tractable algorithms for more related problems.

Mingyu Xiao and Hiroshi Nagamochi. Brief Announcement: Bounded-Degree Cut is Fixed-Parameter Tractable. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 112:1-112:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{xiao_et_al:LIPIcs.ICALP.2018.112, author = {Xiao, Mingyu and Nagamochi, Hiroshi}, title = {{Brief Announcement: Bounded-Degree Cut is Fixed-Parameter Tractable}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {112:1--112:6}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.112}, URN = {urn:nbn:de:0030-drops-91164}, doi = {10.4230/LIPIcs.ICALP.2018.112}, annote = {Keywords: FPT, Important Cuts, Graph Cuts, Graph Algorithms} }

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**Published in:** LIPIcs, Volume 64, 27th International Symposium on Algorithms and Computation (ISAAC 2016)

In this paper, we study the problem of finding an integral multiflow which maximizes the sum of flow values between every two terminals in an undirected tree with a nonnegative integer edge capacity and a set of terminals. In general, it is known that the flow value of an integral multiflow is bounded by the cut value of a cut-system which consists of disjoint subsets each of which contains exactly one terminal or has an odd cut value, and there exists a pair of an integral multiflow and a cut-system whose flow value and cut value are equal; i.e., a pair of a maximum integral multiflow and a minimum cut. In this paper, we propose an O(n)-time algorithm that finds such a pair of an integral multiflow and a cut-system in a given tree instance with n vertices. This improves the best previous results by a factor of Omega(n). Regarding a given tree in an instance as a rooted tree, we define O(n) rooted tree instances taking each vertex as a root, and establish a recursive formula on maximum integral multiflow values of these instances to design a dynamic programming that computes the maximum integral multiflow values of all O(n) rooted instances in linear time. We can prove that the algorithm implicitly maintains a cut-system so that not only a maximum integral multiflow but also a minimum cut-system can be constructed in linear time for any rooted instance whenever it is necessary. The resulting algorithm is rather compact and succinct.

Mingyu Xiao and Hiroshi Nagamochi. A Linear-Time Algorithm for Integral Multiterminal Flows in Trees. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 62:1-62:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{xiao_et_al:LIPIcs.ISAAC.2016.62, author = {Xiao, Mingyu and Nagamochi, Hiroshi}, title = {{A Linear-Time Algorithm for Integral Multiterminal Flows in Trees}}, booktitle = {27th International Symposium on Algorithms and Computation (ISAAC 2016)}, pages = {62:1--62:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-026-2}, ISSN = {1868-8969}, year = {2016}, volume = {64}, editor = {Hong, Seok-Hee}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2016.62}, URN = {urn:nbn:de:0030-drops-68311}, doi = {10.4230/LIPIcs.ISAAC.2016.62}, annote = {Keywords: Multiterminal flow; Maximum flow; Minimum Cut; Trees; Linear-time algorithms} }

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**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

The Traveling Tournament Problem is a complex combinatorial optimization problem in tournament timetabling, which asks a schedule of home/away games meeting specific feasibility requirements, while also minimizing the total distance traveled by all the n teams (n is even). Despite intensive algorithmic research on this problem over the last decade, most instances with more than 10 teams in well-known benchmarks are still unsolved. In this paper, we give a practical approximation algorithm for the problem with constraints such that at most two consecutive home games or away games are allowed. Our algorithm, that generates feasible schedules based on minimum perfect matchings in the underlying graph, not only improves the previous approximation ratio from (1+16/n) to about (1+4/n) but also has very good experimental performances. By applying our schedules on known benchmark sets, we can beat all previously-known results of instances with n being a multiple of 4 by 3% to 10%.

Mingyu Xiao and Shaowei Kou. An Improved Approximation Algorithm for the Traveling Tournament Problem with Maximum Trip Length Two. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 89:1-89:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{xiao_et_al:LIPIcs.MFCS.2016.89, author = {Xiao, Mingyu and Kou, Shaowei}, title = {{An Improved Approximation Algorithm for the Traveling Tournament Problem with Maximum Trip Length Two}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {89:1--89:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.89}, URN = {urn:nbn:de:0030-drops-64976}, doi = {10.4230/LIPIcs.MFCS.2016.89}, annote = {Keywords: sports scheduling, traveling tournament problem, approximation algorithms, timetabling combinatorial optimization} }

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