Основные понятия
The Independent Stable Set problem seeks to find a stable set of vertices in a graph that is also independent with respect to a given matroid. This problem generalizes several well-studied algorithmic problems, including Rainbow Independent Set, Rainbow Matching, and Bipartite Matching with Separation.
Аннотация
The paper studies the computational complexity of the Independent Stable Set problem. It provides the following key insights:
Unconditional lower bound: When the input matroids are represented by independence oracles, there is no algorithm that can solve Independent Stable Set using f(k) · no(k) oracle calls for any computable function f. This lower bound holds even for bipartite, chordal, claw-free, and AT-free graphs.
Parameterized complexity on sparse graphs:
For d-degenerate graphs, Independent Stable Set is FPT when parameterized by d + k, and admits a polynomial kernel when d is a constant.
However, the problem does not admit a polynomial kernel when parameterized by k + d unless NP ⊆ coNP/poly, even for partition matroids.
For graphs with maximum degree ∆, Independent Stable Set admits a polynomial kernel with a graph of size at most k^2∆.
Chordal graphs and linear matroids:
When the input graph is chordal and the matroid is linear, given by its representation, Independent Stable Set can be solved in 2^O(k) · ||A||^O(1) time by a one-sided error Monte Carlo algorithm.
However, the problem does not admit a polynomial kernel on chordal graphs and partition matroids, unless NP ⊆ coNP/poly.
The paper provides a comprehensive analysis of the computational complexity of Independent Stable Set, establishing both positive and negative results for various graph classes and matroid representations.