The q-Coloring problem asks whether the vertices of a graph can be properly colored with q colors. Lokshtanov et al. [SODA 2011] showed that q-Coloring on graphs with a feedback vertex set of size k cannot be solved in time O * ((q − ε) k ), for any ε > 0, unless the Strong Exponential-Time Hypothesis (SETH) fails. In this paper we perform a fine-grained analysis of the complexity of q-Coloring with respect to a hierarchy of parameters. We show that even when parameterized by the vertex cover number, q must appear in the base of the exponent: Unless ETH fails, there is no universal constant θ such that q-Coloring parameterized by vertex cover can be solved in time O * (θ k ) for all fixed q. We apply a method due to Jansen and Kratsch [Inform. & Comput. 2013] to prove that there are O * ((q − ε) k ) time algorithms where k is the vertex deletion distance to several graph classes F for which q-Coloring is known to be solvable in polynomial time. We generalize earlier ad-hoc results by showing that if F is a class of graphs whose (q + 1)-colorable members have bounded treedepth, then there exists some ε > 0 such that q-Coloring can be solved in time O * ((q − ε) k ) when parameterized by the size of a given modulator to F. In contrast, we prove that if F is the class of paths -some of the simplest graphs of unbounded treedepth -then no such algorithm can exist unless SETH fails. *
In this work, we study the d-Hitting Set and Feedback Vertex Set problems through the paradigm of finding diverse collections of r solutions of size at most k each, which has recently been introduced to the field of parameterized complexity [Baste et al., 2019]. This paradigm is aimed at addressing the loss of important side information which typically occurs during the abstraction process which models real-world problems as computational problems. We use two measures for the diversity of such a collection: the sum of all pairwise Hamming distances, and the minimum pairwise Hamming distance. We show that both problems are FPT in k + r for both diversity measures. A key ingredient in our algorithms is a (problem independent) network flow formulation that, given a set of 'base' solutions, computes a maximally diverse collection of solutions. We believe that this could be of independent interest.
We give a first polynomial-time algorithm for (Weighted) Feedback Vertex Set on graphs of bounded maximum induced matching width (mim-width). Explicitly, given a branch decomposition of mim-width w, we give an n O(w)-time algorithm that solves Feedback Vertex Set. This provides a unified polynomial-time algorithm for many well-known classes, such as Interval graphs, Permutation graphs, and Leaf power graphs (given a leaf root), and furthermore, it gives the first polynomial-time algorithms for other classes of bounded mimwidth, such as Circular Permutation and Circular k-Trapezoid graphs (given a circular k-trapezoid model) for fixed k. We complement our result by showing that Feedback Vertex Set is W[1]-hard when parameterized by w and the hardness holds even when a linear branch decomposition of mim-width w is given.
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