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. *

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When modeling an application of practical relevance as an instance of a combinatorial problem X, we are often interested not merely in finding one optimal solution for that instance, but in finding a sufficiently diverse collection of good solutions. In this work we initiate a systematic study of diversity from the point of view of fixed-parameter tractability theory. We consider an intuitive notion of diversity of a collection of solutions which suits a large variety of combinatorial problems of practical interest. Our main contribution is an algorithmic framework which --automatically-- converts a tree-decomposition-based dynamic programming algorithm for a given combinatorial problem X into a dynamic programming algorithm for the diverse version of X. Surprisingly, our algorithm has a polynomial dependence on the diversity parameter.

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.

We are pleased to dedicate this survey on kernelization of the Vertex Cover problem, to Professor Juraj Hromkovič on the occasion of his 60th birthday. The Vertex Cover problem is often referred to as the Drosophila of parameterized complexity. It enjoys a long history. New and worthy perspectives will always be demonstrated first with concrete results here. This survey discusses several research directions in Vertex Cover kernelization. The Barrier Degree of Vertex Cover is discussed. We have reduction rules that kernelize vertices of small degree, including in this paper new results that reduce graphs almost to minimum degree five. Can this process go on forever? What is the minimum vertex-degree barrier for polynomial-time kernelization? Assuming the Exponential-Time Hypothesis, there is a minimum degree barrier. The idea of automated kernelization is discussed. We here report the first experimental results of an AI-guided branching algorithm for Vertex Cover whose logic seems amenable for application in finding reduction rules to kernelize small-degree vertices. The survey highlights a central open problem in parameterized complexity. Happy Birthday, Juraj! * Michael R. Fellows, Lars Jaffke, Alíz Izabella Király and Frances A. Rosamond acknowledge support from the Bergen Research Foundation (BFS).1 In the textbook [17], the problem was entertainingly introduced as 'Bar Fight Prevention'.

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