The 2026 Summer School on Combinatorics and Algorithms is a venue for students and early-career researchers to learn selected topics in theoretical computer science and discrete mathematics. It will be a great opportunity for young and aspiring researchers to study topics which are important but not covered during the lectures in the university classes.
On April 16, 2024, Magnus Wahlström from Royal Holloway, University of London gave a talk at the Discrete Math Seminar on various algorithmic aspects of linear delta-matroids. The title of his talk was “Algorithmic aspects of linear delta-matroids“.
Delta-matroids are a generalization of matroids with connections to many parts of graph theory and combinatorics (such as matching theory and the structure of topological graph embeddings). Formally, a delta-matroid is a pair $D=(V,\mathcal F)$ where $\mathcal F$ is a collection of subsets of V known as “feasible sets.” (They can be thought of as generalizing the set of bases of a matroid, while relaxing the condition that all bases must have the same cardinality.)
Like with matroids, an important class of delta-matroids are linear delta-matroids, where the feasible sets are represented via a skew-symmetric matrix. Prominent examples of linear delta-matroids include linear matroids and matching delta-matroids (where the latter are represented via the famous Tutte matrix).
However, the study of algorithms over delta-matroids seems to have been much less developed than over matroids.
In this talk, we review recent results on representations of and algorithms over linear delta-matroids. We first focus on classical polynomial-time aspects. We present a new (equivalent) representation of linear delta-matroids that is more suitable for algorithmic purposes, and we show that so-called delta-sums and unions of linear delta-matroids are linear. As a result, we get faster (randomized) algorithms for Linear Delta-matroid Parity and Linear Delta-matroid Intersection, improving results from Geelen et al. (2004).
We then move on to parameterized complexity aspects of linear delta-matroids. We find that many results regarding linear matroids which have had applications in FPT algorithms and kernelization directly generalize to linear delta-matroids of bounded rank. On the other hand, unlike with matroids, there is a significant difference between the “rank” and “cardinality” parameters – the structure of bounded-cardinality feasible sets in a delta-matroid of unbounded rank is significantly harder to deal with than feasible sets in a bounded-rank delta-matroid.
