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Odd-Cycle-Packing-Treewidth: On the Maximum Independent Set Problem in Odd-Minor-Free Graph Classes

Authors Mujin Choi , Maximilian Gorsky , Gunwoo Kim , Caleb McFarland , Sebastian Wiederrecht

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LIPIcs.ICALP.2026.64.pdf
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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Mathematics of computing → Graphs and surfaces
  • Mathematics of computing → Combinatorial algorithms
  • Mathematics of computing → Combinatorial optimization
  • Theory of computation → Fixed parameter tractability
Keywords
  • Odd-minor
  • treewidth
  • parameterized algorithm
  • graph minor
  • structural graph theory
  • Odd-Cycle-Packing-treewidth
  • Maximum Independent Set problem

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Abstract

We introduce the tree-decomposition-based graph parameter Odd-Cycle-Packing-treewidth (OCP-tw) as a width parameter that asks to decompose a given graph into pieces of bounded odd cycle packing number. The parameter OCP-tw is monotone under the odd-minor-relation and we provide an analogue to the celebrated Grid Theorem of Robertson and Seymour for OCP-tw. That is, we identify two infinite families of grid-like graphs whose presence as odd-minors implies large OCP-tw and prove that their absence implies bounded OCP-tw. This structural result is constructive and implies a 2^poly(k) poly(n)-time parameterized poly(k)-approximation algorithm for OCP-tw.
Moreover, we show that the (weighted) Maximum Independent Set problem (MIS) can be solved in polynomial time on graphs of bounded OCP-tw. Finally, we lift the concept of OCP-tw to a parameter for matrices of integer programs. To this end, we show that our strategy can be applied to efficiently solve integer programs whose matrices have entries in {-1,0,1} and can be "tree-decomposed" into totally Δ-modular matrices with at most two non-zero entries per row.

Cite As Get BibTex

Mujin Choi, Maximilian Gorsky, Gunwoo Kim, Caleb McFarland, and Sebastian Wiederrecht. Odd-Cycle-Packing-Treewidth: On the Maximum Independent Set Problem in Odd-Minor-Free Graph Classes. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 64:1-64:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.64

Author Details

Mujin Choi
  • Department of Mathematical Sciences, KAIST, Daejeon, South Korea
  • Discrete Mathematics Group, Institute for Basic Science (IBS), Daejeon, South Korea
Maximilian Gorsky
  • Discrete Mathematics Group, Institute for Basic Science (IBS), Daejeon, South Korea
Gunwoo Kim
  • School of Computing, KAIST, Daejeon, South Korea
  • Discrete Mathematics Group, Institute for Basic Science (IBS), Daejeon, South Korea
Caleb McFarland
  • School of Mathematics, Georgia Institute of Technology, Atlanta, GA, USA
Sebastian Wiederrecht
  • School of Computing, KAIST, Daejeon, South Korea

Funding

  • Choi, Mujin: Supported by the Institute for Basic Science (IBS-R029-C1).
  • Gorsky, Maximilian: Supported by the Institute for Basic Science (IBS-R029-C1).
  • Kim, Gunwoo: Supported by the Institute for Basic Science (IBS-R029-C1) and the National Research Foundation, Korea (RS-2025- 00563533).

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