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DTSTART:20230101T000000
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DTSTART;TZID=Asia/Seoul:20240416T163000
DTEND;TZID=Asia/Seoul:20240416T173000
DTSTAMP:20260418T041524
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SUMMARY:Magnus Wahlström\, Algorithmic aspects of linear delta-matroids
DESCRIPTION: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.) \nLike 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). \nHowever\, the study of algorithms over delta-matroids seems to have been much less developed than over matroids. \nIn 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). \nWe 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.
URL:https://dimag.ibs.re.kr/event/2024-04-16/
LOCATION:Room B332\, IBS (기초과학연구원)
CATEGORIES:Discrete Math Seminar
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