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Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.29
A Tight Monte-Carlo Algorithm for Steiner Tree Parameterized by Clique-Width

Authors: Narek Bojikian and Stefan Kratsch

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

Abstract
Given a graph G = (V,E), a set T ⊆ V, and an integer b, the Steiner Tree problem asks whether G has a connected subgraph H with at most b vertices that spans all of T. This work presents a 3^k⋅ n^𝒪(1) time one-sided Monte-Carlo algorithm for solving Steiner Tree when additionally a clique-expression of width k is provided. Known lower bounds for less expressive parameters imply that this dependence on the clique-width of G is optimal assuming the Strong Exponential-Time Hypothesis (SETH). Indeed our work establishes that the parameter dependence of Steiner Tree is the same for any graph parameter between cutwidth and clique-width, assuming SETH. Our work contributes to the program of determining the exact parameterized complexity of fundamental hard problems relative to structural graph parameters such as treewidth, which was initiated by Lokshtanov et al. [SODA 2011 & TALG 2018] and which by now has seen a plethora of results. Since the cut-and-count framework of Cygan et al. [FOCS 2011 & TALG 2022], connectivity problems have played a key role in this program as they pose many challenges for developing tight upper and lower bounds. Recently, Hegerfeld and Kratsch [ESA 2023] gave the first application of the cut-and-count technique to problems parameterized by clique-width and obtained tight bounds for Connected Dominating Set and Connected Vertex Cover, leaving open the complexity of other benchmark connectivity problems such as Steiner Tree and Feedback Vertex Set. Our algorithm for Steiner Tree does not follow the cut-and-count technique and instead works with the connectivity patterns of partial solutions. As a first technical contribution we identify a special family of so-called complete patterns that has strong (existential) representation properties, and using these at least one solution will be preserved. Furthermore, there is a family of 3^k basis patterns that (parity) represents the complete patterns, i.e., it has the same number of solutions modulo two. Our main technical contribution, a new technique called "isolating a representative," allows us to leverage both forms of representation (existential and parity). Both complete patterns and isolation of a representative will likely be applicable to other (connectivity) problems.
Cite as

Narek Bojikian and Stefan Kratsch. A Tight Monte-Carlo Algorithm for Steiner Tree Parameterized by Clique-Width. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 29:1-29:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bojikian_et_al:LIPIcs.ICALP.2024.29,
  author =	{Bojikian, Narek and Kratsch, Stefan},
  title =	{{A Tight Monte-Carlo Algorithm for Steiner Tree Parameterized by Clique-Width}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{29:1--29:18},
  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.2024.29},
  URN =		{urn:nbn:de:0030-drops-201728},
  doi =		{10.4230/LIPIcs.ICALP.2024.29},
  annote =	{Keywords: Parameterized complexity, Steiner tree, clique-width}
}
Document
DOI: 10.4230/LIPIcs.STACS.2023.14
Tight Bounds for Connectivity Problems Parameterized by Cutwidth

Authors: Narek Bojikian, Vera Chekan, Falko Hegerfeld, and Stefan Kratsch

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

Abstract
In this work we start the investigation of tight complexity bounds for connectivity problems parameterized by cutwidth assuming the Strong Exponential-Time Hypothesis (SETH). Van Geffen et al. [Bas A. M. van Geffen et al., 2020] posed this question for Odd Cycle Transversal and Feedback Vertex Set. We answer it for these two and four further problems, namely Connected Vertex Cover, Connected Dominating Set, Steiner Tree, and Connected Odd Cycle Transversal. For the latter two problems it sufficed to prove lower bounds that match the running time inherited from parameterization by treewidth; for the others we provide faster algorithms than relative to treewidth and prove matching lower bounds. For upper bounds we first extend the idea of Groenland et al. [Carla Groenland et al., 2022] to solve what we call coloring-like problems. Such problems are defined by a symmetric matrix M over 𝔽₂ indexed by a set of colors. The goal is to count the number (modulo some prime p) of colorings of a graph such that M has a 1-entry if indexed by the colors of the end-points of any edge. We show that this problem can be solved faster if M has small rank over 𝔽_p. We apply this result to get our upper bounds for CVC and CDS. The upper bounds for OCT and FVS use a subdivision trick to get below the bounds that matrix rank would yield.
Cite as

Narek Bojikian, Vera Chekan, Falko Hegerfeld, and Stefan Kratsch. Tight Bounds for Connectivity Problems Parameterized by Cutwidth. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 14:1-14:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bojikian_et_al:LIPIcs.STACS.2023.14,
  author =	{Bojikian, Narek and Chekan, Vera and Hegerfeld, Falko and Kratsch, Stefan},
  title =	{{Tight Bounds for Connectivity Problems Parameterized by Cutwidth}},
  booktitle =	{40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)},
  pages =	{14:1--14:16},
  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.2023.14},
  URN =		{urn:nbn:de:0030-drops-176667},
  doi =		{10.4230/LIPIcs.STACS.2023.14},
  annote =	{Keywords: Parameterized complexity, connectivity problems, cutwidth}
}
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