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Fine-Grained Complexity of Computing Degree-Constrained Spanning Trees

Authors Narek Bojikian , Alexander Firbas , Robert Ganian , Hung P. Hoang , Krisztina Szilágyi

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LIPIcs.ICALP.2026.38.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Parameterized complexity
  • Structural parameters
  • Clique-width
  • fine-grained complexity
  • Spanning tree

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Abstract

We investigate the computation of minimum-cost spanning trees satisfying prescribed vertex degree constraints: Given a graph G and a constraint function D, we ask for a (minimum-cost) spanning tree T such that for each vertex v, T achieves a degree specified by D(v). Specifically, we consider three kinds of constraint functions ordered by their generality - D may either assign to each vertex a list of admissible degrees, an upper bound on the degree, or a specific degree. Using a combination of novel techniques and state-of-the-art machinery, we obtain an almost-complete overview of the fine-grained complexity of these problems taking into account the most classical structural graph parameters of the input graph G. In particular, we present SETH-tight upper and lower bounds for these problems when parameterized by pathwidth and cutwidth, an ETH-tight algorithm parameterized by clique-width, and a nearly SETH-tight algorithm parameterized by treewidth.
In order to obtain our upper bound for clique-width, we develop a novel technique of double representation through "requirement shifting". Using this technique, we also obtain an ETH-tight single-exponential XP algorithm for the Exact Leaf Spanning Tree problem parameterized by clique-width, which settles the final remaining open case for clique-width from the classical Cut and Count of Cygan et al. [FOCS 2011, TALG 2022]. This shows the versatility of our technique and its potential applicability to other problems as well. Additionally, in order to establish our lower and upper bounds we introduce a number of tools which may be of independent interest, including lazy coloring and "asymptotic" SETH-based reductions for structural parameters.

Cite As Get BibTex

Narek Bojikian, Alexander Firbas, Robert Ganian, Hung P. Hoang, and Krisztina Szilágyi. Fine-Grained Complexity of Computing Degree-Constrained Spanning Trees. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 38:1-38:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.38

Author Details

Narek Bojikian
  • Humboldt Universität zu Berlin, Germany
Alexander Firbas
  • TU Wien, Austria
Robert Ganian
  • TU Wien, Austria
Hung P. Hoang
  • TU Wien, Austria
Krisztina Szilágyi
  • Czech Technical University, Prague, Czech Republic

Funding

  • Firbas, Alexander: FWF Project 10.55776/Y1329 and WWTF Project 10.47379/ICT22029.
  • Ganian, Robert: FWF Project 10.55776/Y1329 and WWTF Project 10.47379/ICT22029.
  • Hoang, Hung P.: FWF Projects 10.55776/Y1329 and ESP1136425.
  • Szilágyi, Krisztina: Supported under the project Robotics and advanced industrial production (reg. no. CZ.02.01.01/00/22_008/0004590) and CTU Global Postdoc Fellowship Program.

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