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.

翻译：我们提出了线性Delta-拟阵（一种推广了拟阵的集合系统，与匹配理论和图嵌入具有重要联系）的代表集风格表述。此外，我们的证明采用了一种新的多项式族筛选方法，该方法将代表集引理的线性代数方法推广到了有界次多项式的设定中。线性Delta-拟阵的代表集表述随后通过分析表示该Delta-拟阵的斜对称矩阵的Pfaffian得出。将相同框架应用于行列式而非Pfaffian，即可恢复线性拟阵的代表集引理。总体而言，这显著扩展了可用于核化的工具集。作为应用，我们展示了Mader网络的一个精确稀疏化结果：设$G=(V,E)$为一个图，$\mathcal{T}$为终端集$T \subseteq V(G)$（$|T|=k$）的一个划分。$G$中的$\mathcal{T}$-路径是指端点位于$\mathcal{T}$不同区块、且内部顶点与$T$不相交的路径。在多项式时间内，我们可以推导出一个图$G'=(V',E')$（满足$T \subseteq V(G')$），使得对于每个子集$S \subseteq T$，$G$中存在端点集为$S$的$\mathcal{T}$-路径当且仅当$G'$中亦存在，且$|V(G')|=O(k^3)$。这推广了（无向版本的）割覆盖引理（对应于$\mathcal{T}$仅包含两个区块的情形）。为证明Mader网络稀疏化结果，我们进一步定义了Mader Delta-拟阵类，并证明它们具有线性表示。这一点本身应具有独立的研究价值。