3 Search Results for "Boehmer, Niclas"

Document
DOI: 10.4230/LIPIcs.STACS.2023.39
Induced Matching Below Guarantees: Average Paves the Way for Fixed-Parameter Tractability

Authors: Tomohiro Koana

Published in: LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)

Abstract
In this work, we study the Induced Matching problem: Given an undirected graph G and an integer 𝓁, is there an induced matching M of size at least 𝓁? An edge subset M is an induced matching in G if M is a matching such that there is no edge between two distinct edges of M. Our work looks into the parameterized complexity of Induced Matching with respect to "below guarantee" parameterizations. We consider the parameterization u - 𝓁 for an upper bound u on the size of any induced matching. For instance, any induced matching is of size at most n/2 where n is the number of vertices, which gives us a parameter n/2 - 𝓁. In fact, there is a straightforward 9^{n/2 - 𝓁} ⋅ n^O(1)-time algorithm for Induced Matching [Moser and Thilikos, J. Discrete Algorithms]. Motivated by this, we ask: Is Induced Matching FPT for a parameter smaller than n/2 - 𝓁? In search for such parameters, we consider MM(G) - 𝓁 and IS(G) - 𝓁, where MM(G) is the maximum matching size and IS(G) is the maximum independent set size of G. We find that Induced Matching is presumably not FPT when parameterized by MM(G) - 𝓁 or IS(G) - 𝓁. In contrast to these intractability results, we find that taking the average of the two helps - our main result is a branching algorithm that solves Induced Matching in 49^{(MM(G) + IS(G))/ 2 - 𝓁} ⋅ n^O(1) time. Our algorithm makes use of the Gallai-Edmonds decomposition to find a structure to branch on.
Cite as

Tomohiro Koana. Induced Matching Below Guarantees: Average Paves the Way for Fixed-Parameter Tractability. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 39:1-39:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{koana:LIPIcs.STACS.2023.39,
  author =	{Koana, Tomohiro},
  title =	{{Induced Matching Below Guarantees: Average Paves the Way for Fixed-Parameter Tractability}},
  booktitle =	{40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)},
  pages =	{39:1--39:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-266-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{254},
  editor =	{Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.39},
  URN =		{urn:nbn:de:0030-drops-176914},
  doi =		{10.4230/LIPIcs.STACS.2023.39},
  annote =	{Keywords: Parameterized Complexity, Below Guarantees, Induced Matching, Gallai-Edmonds Decomposition}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2022.21
Deepening the (Parameterized) Complexity Analysis of Incremental Stable Matching Problems

Authors: Niclas Boehmer, Klaus Heeger, and Rolf Niedermeier

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

Abstract
When computing stable matchings, it is usually assumed that the preferences of the agents in the matching market are fixed. However, in many realistic scenarios, preferences change over time. Consequently, an initially stable matching may become unstable. Then, a natural goal is to find a matching which is stable with respect to the modified preferences and as close as possible to the initial one. For Stable Marriage/Roommates, this problem was formally defined as Incremental Stable Marriage/Roommates by Bredereck et al. [AAAI '20]. As they showed that Incremental Stable Roommates and Incremental Stable Marriage with Ties are NP-hard, we focus on the parameterized complexity of these problems. We answer two open questions of Bredereck et al. [AAAI '20]: We show that Incremental Stable Roommates is W[1]-hard parameterized by the number of changes in the preferences, yet admits an intricate XP-algorithm, and we show that Incremental Stable Marriage with Ties is W[1]-hard parameterized by the number of ties. Furthermore, we analyze the influence of the degree of "similarity" between the agents' preference lists, identifying several polynomial-time solvable and fixed-parameter tractable cases, but also proving that Incremental Stable Roommates and Incremental Stable Marriage with Ties parameterized by the number of different preference lists are W[1]-hard.
Cite as

Niclas Boehmer, Klaus Heeger, and Rolf Niedermeier. Deepening the (Parameterized) Complexity Analysis of Incremental Stable Matching Problems. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 21:1-21:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{boehmer_et_al:LIPIcs.MFCS.2022.21,
  author =	{Boehmer, Niclas and Heeger, Klaus and Niedermeier, Rolf},
  title =	{{Deepening the (Parameterized) Complexity Analysis of Incremental Stable Matching Problems}},
  booktitle =	{47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
  pages =	{21:1--21:15},
  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.2022.21},
  URN =		{urn:nbn:de:0030-drops-168194},
  doi =		{10.4230/LIPIcs.MFCS.2022.21},
  annote =	{Keywords: Stable Marriage, Stable Roommates, adapting to changing preferences, NP-hardness, W\lbrack1\rbrack-hardness, XP, FPT, master lists, incremental algorithms}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2022.27
The Complexity of Finding Fair Many-To-One Matchings

Authors: Niclas Boehmer and Tomohiro Koana

Published in: LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Abstract
We analyze the (parameterized) computational complexity of "fair" variants of bipartite many-to-one matching, where each vertex from the "left" side is matched to exactly one vertex and each vertex from the "right" side may be matched to multiple vertices. We want to find a "fair" matching, in which each vertex from the right side is matched to a "fair" set of vertices. Assuming that each vertex from the left side has one color modeling its attribute, we study two fairness criteria. In one of them, we deem a vertex set fair if for any two colors, the difference between the numbers of their occurrences does not exceed a given threshold. Fairness is relevant when finding many-to-one matchings between students and colleges, voters and constituencies, and applicants and firms. Here colors may model sociodemographic attributes, party memberships, and qualifications, respectively. We show that finding a fair many-to-one matching is NP-hard even for three colors and maximum degree five. Our main contribution is the design of fixed-parameter tractable algorithms with respect to the number of vertices on the right side. Our algorithms make use of a variety of techniques including color coding. At the core lie integer linear programs encoding Hall like conditions. To establish the correctness of our integer programs, we prove a new separation result, inspired by Frank’s separation theorem [Frank, Discrete Math. 1982], which may also be of independent interest. We further obtain complete complexity dichotomies regarding the number of colors and the maximum degree of each side.
Cite as

Niclas Boehmer and Tomohiro Koana. The Complexity of Finding Fair Many-To-One Matchings. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 27:1-27:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{boehmer_et_al:LIPIcs.ICALP.2022.27,
  author =	{Boehmer, Niclas and Koana, Tomohiro},
  title =	{{The Complexity of Finding Fair Many-To-One Matchings}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{27:1--27:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.27},
  URN =		{urn:nbn:de:0030-drops-163680},
  doi =		{10.4230/LIPIcs.ICALP.2022.27},
  annote =	{Keywords: Graph theory, polynomial-time algorithms, NP-hardness, FPT, ILP, color coding, submodular and supermodular functions, algorithmic fairness}
}
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