We study the following Independent Stable Set problem. Let G be an undirected graph and M = (V(G),I) be a matroid whose elements are the vertices of G. For an integer k\geq 1, the task is to decide whether G contains a set S\subseteq V(G) of size at least k which is independent (stable) in G and independent in M. This problem generalizes several well-studied algorithmic problems, including Rainbow Independent Set, Rainbow Matching, and Bipartite Matching with Separation. We show that - When the matroid M is represented by the independence oracle, then for any computable function f, no algorithm can solve Independent Stable Set using f(k)n^{o(k)} calls to the oracle. - On the other hand, when the graph G is of degeneracy d, then the problem is solvable in time O((d+1)^kn), and hence is FPT parameterized by d+k. Moreover, when the degeneracy d is a constant (which is not a part of the input), the problem admits a kernel polynomial in k. More precisely, we prove that for every integer d\geq 0, the problem admits a kernelization algorithm that in time n^{O(d)} outputs an equivalent framework with a graph on dk^{O(d)} vertices. A lower bound complements this when d is part of the input: Independent Stable Set does not admit a polynomial kernel when parameterized by k+d unless NP \subseteq coNP/poly. This lower bound holds even when M is a partition matroid. - Another set of results concerns the scenario when the graph G is chordal. In this case, our computational lower bound excludes an FPT algorithm when the input matroid is given by its independence oracle. However, we demonstrate that Independent Stable Set can be solved in 2^{O(k)}||M||^{O(1)} time when M is a linear matroid given by its representation. In the same setting, Independent Stable Set does not have a polynomial kernel when parameterized by k unless NP\subseteq coNP/poly.

翻译：我们研究以下独立稳定集问题。设G为无向图，M=(V(G),I)为以G的顶点为元素的拟阵。对于整数k≥1，任务是判断G中是否存在大小至少为k的子集S⊆V(G)，该子集在G中独立（稳定）且在M中独立。该问题推广了多个经典算法问题，包括彩虹独立集、彩虹匹配和带分离的二分图匹配。我们证明：- 当拟阵M由独立性预言机表示时，对于任意可计算函数f，不存在算法能在f(k)n^{o(k)}次预言机调用内求解独立稳定集。- 另一方面，当图G的退化度为d时，该问题可在O((d+1)^kn)时间内求解，因此对于参数d+k是固定参数可处理的。此外，当退化度d为常数（不包含在输入中）时，该问题存在关于k的多项式核。更精确地，我们证明对任意整数d≥0，该问题在n^{O(d)}时间内可输出一个等价框架，其图包含dk^{O(d)}个顶点。当d作为输入部分时，存在下界与之互补：除非NP⊆coNP/poly，否则参数化为k+d的独立稳定集不存在多项式核。即使M为划分拟阵，该下界依然成立。- 另一组结果涉及图G为弦图的情形。此时，当输入拟阵由其独立性预言机给出时，我们的计算下界排除了固定参数可处理算法。然而我们证明，当M为由其表示给出的线性拟阵时，独立稳定集可在2^{O(k)}||M||^{O(1)}时间内求解。在相同设定下，除非NP⊆coNP/poly，否则参数化为k的独立稳定集不存在多项式核。