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Documents authored by Hörsch, Florian

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Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.86
Problems on Group-Labeled Matroid Bases

Authors: Florian Hörsch, András Imolay, Ryuhei Mizutani, Taihei Oki, and Tamás Schwarcz

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
Consider a matroid equipped with a labeling of its ground set to an abelian group. We define the label of a subset of the ground set as the sum of the labels of its elements. We study a collection of problems on finding bases and common bases of matroids with restrictions on their labels. For zero bases and zero common bases, the results are mostly negative. While finding a non-zero basis of a matroid is not difficult, it turns out that the complexity of finding a non-zero common basis depends on the group. Namely, we show that the problem is hard for a fixed group if it contains an element of order two, otherwise it is polynomially solvable. As a generalization of both zero and non-zero constraints, we further study F-avoiding constraints where we seek a basis or common basis whose label is not in a given set F of forbidden labels. Using algebraic techniques, we give a randomized algorithm for finding an F-avoiding common basis of two matroids represented over the same field for finite groups given as operation tables. The study of F-avoiding bases with groups given as oracles leads to a conjecture stating that whenever an F-avoiding basis exists, an F-avoiding basis can be obtained from an arbitrary basis by exchanging at most |F| elements. We prove the conjecture for the special cases when |F| ≤ 2 or the group is ordered. By relying on structural observations on matroids representable over fixed, finite fields, we verify a relaxed version of the conjecture for these matroids. As a consequence, we obtain a polynomial-time algorithm in these special cases for finding an F-avoiding basis when |F| is fixed.
Cite as

Florian Hörsch, András Imolay, Ryuhei Mizutani, Taihei Oki, and Tamás Schwarcz. Problems on Group-Labeled Matroid Bases. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 86:1-86:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{horsch_et_al:LIPIcs.ICALP.2024.86,
  author =	{H\"{o}rsch, Florian and Imolay, Andr\'{a}s and Mizutani, Ryuhei and Oki, Taihei and Schwarcz, Tam\'{a}s},
  title =	{{Problems on Group-Labeled Matroid Bases}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{86:1--86:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.86},
  URN =		{urn:nbn:de:0030-drops-202299},
  doi =		{10.4230/LIPIcs.ICALP.2024.86},
  annote =	{Keywords: matroids, matroid intersection, congruency constraint, exact-weight constraint, additive combinatorics, algebraic algorithm, strongly base orderability}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2024.57
Multicut Problems in Embedded Graphs: The Dependency of Complexity on the Demand Pattern

Authors: Jacob Focke, Florian Hörsch, Shaohua Li, and Dániel Marx

Published in: LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

Abstract
The Multicut problem asks for a minimum cut separating certain pairs of vertices: formally, given a graph G and a demand graph H on a set T ⊆ V(G) of terminals, the task is to find a minimum-weight set C of edges of G such that whenever two vertices of T are adjacent in H, they are in different components of G⧵ C. Colin de Verdière [Algorithmica, 2017] showed that Multicut with t terminals on a graph G of genus g can be solved in time f(t,g) n^O(√{g²+gt+t}). Cohen-Addad et al. [JACM, 2021] proved a matching lower bound showing that the exponent of n is essentially best possible (for every fixed value of t and g), even in the special case of Multiway Cut, where the demand graph H is a complete graph. However, this lower bound tells us nothing about other special cases of Multicut such as Group 3-Terminal Cut (where three groups of terminals need to be separated from each other). We show that if the demand pattern is, in some sense, close to being a complete bipartite graph, then Multicut can be solved faster than f(t,g) n^{O(√{g²+gt+t})}, and furthermore this is the only property that allows such an improvement. Formally, for a class ℋ of graphs, Multicut(ℋ) is the special case where the demand graph H is in ℋ. For every fixed class ℋ (satisfying some mild closure property), fixed g, and fixed t, our main result gives tight upper and lower bounds on the exponent of n in algorithms solving Multicut(ℋ). In addition, we investigate a similar setting where, instead of parameterizing by the genus g of G, we parameterize by the minimum number k of edges of G that need to be deleted to obtain a planar graph. Interestingly, in this setting it makes a significant difference whether the graph G is weighted or unweighted: further nontrivial algorithmic techniques give substantial improvements in the unweighted case.
Cite as

Jacob Focke, Florian Hörsch, Shaohua Li, and Dániel Marx. Multicut Problems in Embedded Graphs: The Dependency of Complexity on the Demand Pattern. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 57:1-57:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{focke_et_al:LIPIcs.SoCG.2024.57,
  author =	{Focke, Jacob and H\"{o}rsch, Florian and Li, Shaohua and Marx, D\'{a}niel},
  title =	{{Multicut Problems in Embedded Graphs: The Dependency of Complexity on the Demand Pattern}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{57:1--57:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-316-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{293},
  editor =	{Mulzer, Wolfgang and Phillips, Jeff M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.57},
  URN =		{urn:nbn:de:0030-drops-200021},
  doi =		{10.4230/LIPIcs.SoCG.2024.57},
  annote =	{Keywords: MultiCut, Multiway Cut, Parameterized Complexity, Tight Bounds, Embedded Graph, Planar Graph, Genus, Surface, Exponential Time Hypothesis}
}
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