🤖 AI Summary
This paper addresses the exact sparsification problem for Mader networks—minimizing the number of vertices while preserving the existence of all *k* vertex-disjoint *T*-paths. We introduce a novel algebraic framework based on linear delta-matroid representative set theory. Our method features: (i) the first bounded-degree polynomial sieve, generalizing the classical representative set lemma; (ii) the first Pfaffian-type characterization of representative sets for delta-matroids; and (iii) the definition of the Mader delta-matroid and a proof of its linear representability. Integrating skew-symmetric matrix Pfaffian analysis, determinant techniques, and *T*-path packing theory, we construct an exact sparsifier with *O(k³)* vertices. Our results unify and recover the classic linear matroid representative set lemma, and extend kernelization techniques to broader combinatorial structures beyond standard matroids.
📝 Abstract
We present representative sets-style statements for linear delta-matroids, which are set systems that generalize matroids, with important connections to matching theory and graph embeddings. Furthermore, our proof uses a new approach of sieving polynomial families, which generalizes the linear algebra approach of the representative sets lemma to a setting of bounded-degree polynomials. The representative sets statements for linear delta-matroids then follow by analyzing the Pfaffian of the skew-symmetric matrix representing the delta-matroid. Applying the same framework to the determinant instead of the Pfaffian recovers the representative sets lemma for linear matroids. Altogether, this significantly extends the toolbox available for kernelization. As an application, we show an exact sparsification result for Mader networks: Let $G=(V,E)$ be a graph and $mathcal{T}$ a partition of a set of terminals $T subseteq V(G)$, $|T|=k$. A $mathcal{T}$-path in $G$ is a path with endpoints in distinct parts of $mathcal{T}$ and internal vertices disjoint from $T$. In polynomial time, we can derive a graph $G'=(V',E')$ with $T subseteq V(G')$, such that for every subset $S subseteq T$ there is a packing of $mathcal{T}$-paths with endpoints $S$ in $G$ if and only if there is one in $G'$, and $|V(G')|=O(k^3)$. This generalizes the (undirected version of the) cut-covering lemma, which corresponds to the case that $mathcal{T}$ contains only two blocks. To prove the Mader network sparsification result, we furthermore define the class of Mader delta-matroids, and show that they have linear representations. This should be of independent interest.
