Quick Separation in Chordal and Split Graphs

In this paper we study two classical cut problems, namely Multicut and Multiway Cut on chordal graphs and split graphs. In the Multicut problem, the input is a graph G, a collection of vertex pairs (si, ti), i ∈ [], and a positive integer k and the goal is to decide if there exists a vertex subset S ⊆ V (G) \ {si, ti : i ∈ []} of size at most k such that for every vertex pair (si, ti), si and ti are in two different connected components of G − S. In Unrestricted Multicut, the solution S can possibly pick the vertices in the vertex pairs {(si, ti) : i ∈ []}. An important special case of the Multicut problem is the Multiway Cut problem, where instead of vertex pairs, we are given a set T of terminal vertices, and the goal is to separate every pair of distinct vertices in T × T. The fixed parameter tractability (FPT) of these problems was a long-standing open problem and has been resolved fairly recently. Multicut and Multiway Cut now admit algorithms with running times 2O(k3)nO(1) and 2knO(1), respectively. However, the kernelization complexity of both these problemsis not fully resolved: while Multicut cannot admit a polynomial kernel under reasonable complexity assumptions, it is a well known open problem to construct a polynomial kernel for Multiway Cut. Towards designing faster FPT algorithms and polynomial kernels for the above mentioned problems, we study them on chordal and split graphs. In particular we obtain the following results. 1. Multicut on chordal graphs admits a polynomial kernel with O(k37) vertices. Multiway Cuton chordal graphs admits a polynomial kernel with O(k13) vertices. 2. Multicut on chordal graphs can be solved in time min{O(2k·(k3 + )·(n + m)), 2O( log k)·(n +m) + (n + m)}. Hence Multicut on chordal graphs parameterized by the number of terminals is in XP. 3. Multicut on split graphs can be solved in time min{O(1.2738k+kn+(n+m), O(2··(n+m))}. Unrestricted Multicut on split graphs can be solved in time O(4· · (n + m)).publishedVersio

Representative set statements for delta-matroids and the Mader delta-matroid

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