Representative set statements for delta-matroids and the Mader delta-matroid
Magnus Wahlström
TL;DR
This paper extends the algebraic toolkit for kernelization by developing bounded-degree sieving polynomials that generalize the representative-sets technique from linear matroids to linear delta-matroids. It introduces Mader delta-matroids, proves their linear representability, and leverages Pfaffian-based sieving to obtain delta-matroid representative sets with size bounds of $O(k^q)$ for a terminal set of size $k$ and constant $q$. A key application is the Mader-mimicking network, yielding $O(k^3)$-vertex sparsifiers that preserve all relevant S-path packings and cuts, thus generalizing cut-covering results to multi-block terminals. The framework unifies matroid and polynomial approaches to kernelization, enabling new sparsification and network-reduction techniques with potential impact on polynomial kernel lower bounds and related parameterized problems.
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.