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FPT Algorithms over Linear Delta-Matroids with Applications

Authors Eduard Eiben , Tomohiro Koana , Magnus Wahlström

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LIPIcs.ESA.2026.89.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Mathematics of computing → Matroids and greedoids
Keywords
  • Delta-matroids
  • parameterized algorithms
  • triangle packing

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Abstract

Matroids, particularly linear matroids, have been a powerful tool for applications in parameterized complexity, both for algorithms and kernelization. In particular, they have been instrumental in speeding up or replacing dynamic programming. Delta-matroids are a generalization of matroids that further encapsulate structures such as non-maximum matchings in general graphs and various path-packing and topological structures. There is also a notion of linear delta-matroids (represented by skew-symmetric matrices) which carries significant expressive power and enables powerful algorithms. We investigate parameterized complexity aspects of problems defined over linear delta-matroids, or with delta-matroid constraints. Our initial analysis of basic intersection and packing problems reveals a different complexity landscape compared to the more familiar matroid case. In particular, there is a stark contrast in complexity between the cardinality parameter k and the rank parameter r. For example, finding an intersection of size k of three linear delta-matroids is W[1]-hard when parameterized by k, while far more general problems (such as finding a set packing of size k that is feasible in a given linear delta-matroid) are FPT when parameterized by the rank r of the delta-matroid. In fact, we extend the recent determinantal sieving procedure of Eiben, Koana, and Wahlström (TheoretiCS 2025) into a process that sieves a given polynomial for a monomial whose support is feasible in a given linear delta-matroid, parameterized by r. This is a direct generalization of determinantal sieving. 
Second, we investigate a curious class of problems that turns out to be FPT parameterized by k, even on delta-matroids of unbounded rank. We begin with Delta-matroid Triangle Cover - find a feasible set of size k that can be covered by a vertex-disjoint packing of triangles (i.e., sets of size 3) out of a given triangle collection. For example, this allows us to find, in a graph, a packing of K₃’s and K₂’s with the maximum possible number of edges, parameterized above the matching number of the graph (note that this problem is NP-hard, unlike the problem of finding such a packing covering a maximum number of vertices, which is tractable). As applications, we resolve the FPT status of Cluster Subgraph and Strong Triadic Closure parameterized above the matching number.

Cite As Get BibTex

Eduard Eiben, Tomohiro Koana, and Magnus Wahlström. FPT Algorithms over Linear Delta-Matroids with Applications. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 89:1-89:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.89

Author Details

Eduard Eiben
  • Dept. of Computer Science, Royal Holloway, University of London, UK
Tomohiro Koana
  • Graduate School of Information Science and Technology, The University of Tokyo, Japan
Magnus Wahlström
  • Dept. of Computer Science, Royal Holloway, University of London, UK

Funding

  • Eiben, Eduard: Supported by Engineering and Physical Sciences Research Council (EPSRC) grant UKRI4530: Exploring Parameterized Complexity in Blockchain Systems.
  • Koana, Tomohiro: Supported by JST CREST Grant Number JPMJCR24Q2 and JST ERATO Grant Number JPMJER2301.

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