Stability in Graphs with Matroid Constraints
Fedor V. Fomin, Petr A. Golovach, Tuukka Korhonen, Saket Saurabh
TL;DR
The paper studies Independent Stable Set in frameworks (G, M), unifying stable-set and matroid constraints, and shows both hardness and tractability results. An unconditional oracle-based lower bound rules out subexponential oracle-query algorithms for independence-oracle access, while a suite of positive results makes the problem FPT on $d$-degenerate graphs with time $O((d+1)^k\cdot n)$ and yields kernels of size $dk^{O(d)}$ for fixed $d$, and $k^2\Delta$ for bounded degree. For chordal graphs, a representable-matroid setting yields $2^{O(k)}\|A\|^{O(1)}$-time fixed-parameter algorithms via representative sets, alongside a polynomial-kernel barrier for partition matroids. The results collectively map the boundary between intractability and fixed-parameter tractability across graph classes and matroid access models, providing a versatile framework for Rainbow-type problems and related combinatorial optimization tasks.
Abstract
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.