A kernelization is an efficient algorithm that given an instance of a parameterized problem returns an equivalent instance of size bounded by some function of the input parameter value. It is quite well understood which problems do or (conditionally) do not admit a kernelization where this size bound is polynomial, a so-called polynomial kernelization. Unfortunately, such polynomial kernelizations are known only in fairly restrictive settings where a small parameter value corresponds to a strong restriction on the global structure on the instance. Motivated by this, Antipov and Kratsch [WG 2025] proposed a local variant of kernelization, called boundaried kernelization, that requires only local structure to achieve a local improvement of the instance, which is in the spirit of protrusion replacement used in meta-kernelization [Bodlaender et al.\ JACM 2016]. They obtain polynomial boundaried kernelizations as well as (unconditional) lower bounds for several well-studied problems in kernelization. In this work, we leverage the matroid-based techniques of Kratsch and Wahlstr\"om [JACM 2020] to obtain randomized polynomial boundaried kernelizations for \smultiwaycut, \dtmultiwaycut, \oddcycletransversal, and \vertexcoveroct, for which randomized polynomial kernelizations in the usual sense were known before. A priori, these techniques rely on the global connectivity of the graph to identify reducible (irrelevant) vertices. Nevertheless, the separation of the local part by its boundary turns out to be sufficient for a local application of these methods.

翻译：核化是一种高效算法，它接收参数化问题的实例，并返回一个规模受输入参数值函数约束的等价实例。关于哪些问题存在或（有条件地）不存在多项式规模约束的核化（即多项式核化），已有相当深入的理解。遗憾的是，这类多项式核化仅在相当受限的场景中已知，其中较小的参数值对应于实例全局结构的强约束。受此启发，Antipov与Kratsch [WG 2025] 提出了一种局部核化变体，称为边界核化，它仅需局部结构即可实现实例的局部改进，其精神与元核化中使用的突起替换技术 [Bodlaender等人, JACM 2016] 一脉相承。他们为核化领域中多个深入研究的问题获得了多项式边界核化以及（无条件）下界。在本工作中，我们利用Kratsch与Wahlström [JACM 2020] 提出的基于拟阵的技术，为\smultiwaycut、\dtmultiwaycut、\oddcycletransversal及\vertexcoveroct问题构建了随机化多项式边界核化算法，而这些问题在传统意义上已存在随机化多项式核化。从原理上看，这些技术依赖于图的全局连通性来识别可约（无关）顶点。然而，局部部分通过其边界的分离被证明足以支持这些方法的局部化应用。