A framework consists of an undirected graph $G$ and a matroid $M$ whose elements correspond to the vertices of $G$. Recently, Fomin et al. [SODA 2023] and Eiben et al. [ArXiV 2023] developed parameterized algorithms for computing paths of rank $k$ in frameworks. More precisely, for vertices $s$ and $t$ of $G$, and an integer $k$, they gave FPT algorithms parameterized by $k$ deciding whether there is an $(s,t)$-path in $G$ whose vertex set contains a subset of elements of $M$ of rank $k$. These algorithms are based on Schwartz-Zippel lemma for polynomial identity testing and thus are randomized, and therefore the existence of a deterministic FPT algorithm for this problem remains open. We present the first deterministic FPT algorithm that solves the problem in frameworks whose underlying graph $G$ is planar. While the running time of our algorithm is worse than the running times of the recent randomized algorithms, our algorithm works on more general classes of matroids. In particular, this is the first FPT algorithm for the case when matroid $M$ is represented over rationals. Our main technical contribution is the nontrivial adaptation of the classic irrelevant vertex technique to frameworks to reduce the given instance to one of bounded treewidth. This allows us to employ the toolbox of representative sets to design a dynamic programming procedure solving the problem efficiently on instances of bounded treewidth.

翻译：框架由无向图 $G$ 和拟阵 $M$ 组成，其中 $M$ 的元素对应于 $G$ 的顶点。最近，Fomin 等人 [SODA 2023] 和 Eiben 等人 [ArXiV 2023] 开发了参数化算法，用于计算框架中秩为 $k$ 的路径。更精确地说，对于 $G$ 的顶点 $s$ 和 $t$ 以及整数 $k$，他们给出了参数化为 $k$ 的 FPT 算法，用于判定 $G$ 中是否存在一条 $(s,t)$-路径，其顶点集包含 $M$ 中秩为 $k$ 的元素子集。这些算法基于用于多项式恒等检验的 Schwartz-Zippel 引理，因此是随机化的，因此该问题的确定性 FPT 算法的存在性仍然未知。我们提出了首个解决该问题的确定性 FPT 算法，适用于底层图 $G$ 为平面的框架。虽然我们算法的运行时间逊于近期随机化算法的运行时间，但我们的算法适用于更广泛的拟阵类别。特别地，这是首个针对拟阵 $M$ 在有理数上表示的情形下的 FPT 算法。我们的主要技术贡献是将经典的不相关顶点技术非平凡地适配于框架，从而将给定实例归约为有界树宽的实例。这使得我们能够利用代表集工具箱设计动态规划过程，在有界树宽的实例上高效求解该问题。