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It has long been known that Feedback Vertex Set can be solved in time 2 O (w log w) n O (1) on n-vertex graphs of treewidth w, but it was only recently that this running time was improved to 2 O (w) n O (1) , that is, to single-exponential parameterized by treewidth. We investigate which generalizations of Feedback Vertex Set can be solved in a similar running time. Formally, for a class P of graphs, the Bounded P -Block Vertex Deletion problem asks, given a graph G on n vertices and positive integers k and d, whether G contains a set S of at most k vertices such that each block of G - S has at most d vertices and is in P . Assuming that P is recognizable in polynomial time and satisfies a certain natural hereditary condition, we give a sharp characterization of when single-exponential parameterized algorithms are possible for fixed values of d: if P consists only of chordal graphs, then the problem can be solved in time 2 O (w d 2) n O (1) , if P contains a graph with an induced cycle of length ℓ ⩾ 4 , then the problem is not solvable in time 2 o (w log w) n O (1) even for fixed d = ℓ , unless the ETH fails. We also study a similar problem, called Bounded P -Component Vertex Deletion, where the target graphs have connected components of small size rather than blocks of small size, and we present analogous results. For this problem, we also show that if d is part of the input and P contains all chordal graphs, then it cannot be solved in time f (w) n o (w) for some function f, unless the ETH fails. [ABSTRACT FROM AUTHOR]

In the TOKEN SWAPPING problem we are given a graph with a token placed on each vertex. Each token has exactly one destination vertex, and we try to move all the tokens to their destinations, using the minimum number of swaps, i.e., operations of exchanging the tokens on two adjacent vertices. As the main result of this paper, we show that TOKEN SWAPPING is W[1]-hard parameterized by the length k of a shortest sequence of swaps. In fact, we prove that, for any computable function f, it cannot be solved in time f(k)no(k/logk) where n is the number of vertices of the input graph, unless the ETH fails. This lower bound almost matches the trivial nO(k)-time algorithm. We also consider two generalizations of the TOKEN SWAPPING, namely COLORED TOKEN SWAPPING (where the tokens have colors and tokens of the same color are indistinguishable), and SUBSET TOKEN SWAPPING (where each token has a set of possible destinations). To complement the hardness result, we prove that even the most general variant, SUBSET TOKEN SWAPPING, is FPT in nowhere-dense graph classes. Finally, we consider the complexities of all three problems in very restricted classes of graphs: graphs of bounded treewidth and diameter, stars, cliques, and paths, trying to identify the borderlines between polynomial and NP-hard cases. [ABSTRACT FROM AUTHOR]

The Grundy number of a graph is the maximum number of colors used by the greedy coloring algorithm over all vertex orderings. In this paper, we study the computational complexity of Grundy Coloring , the problem of determining whether a given graph has Grundy number at least k . We also study the variants Weak Grundy Coloring (where the coloring is not necessarily proper) and Connected Grundy Coloring (where at each step of the greedy coloring algorithm, the subgraph induced by the colored vertices must be connected). We show that Grundy Coloring can be solved in time O ∗ ( 2 . 44 3 n ) and Weak Grundy Coloring in time O ∗ ( 2 . 71 6 n ) on graphs of order n . While Grundy Coloring and Weak Grundy Coloring are known to be solvable in time O ∗ ( 2 O ( w k ) ) for graphs of treewidth w (where k is the number of colors), we prove that under the Exponential Time Hypothesis (ETH), they cannot be solved in time O ∗ ( 2 o ( w log w ) ) . We also describe an O ∗ ( 2 2 O ( k ) ) algorithm for Weak Grundy Coloring , which is therefore FPT for the parameter k . Moreover, under the ETH, we prove that such a running time is essentially optimal (this lower bound also holds for Grundy Coloring ). Although we do not know whether Grundy Coloring is in FPT , we show that this is the case for graphs belonging to a number of standard graph classes including chordal graphs, claw-free graphs, and graphs excluding a fixed minor. We also describe a quasi-polynomial time algorithm for Grundy Coloring and Weak Grundy Coloring on apex-minor graphs. In stark contrast with the two other problems, we show that Connected Grundy Coloring is NP -complete already for k = 7 colors. [ABSTRACT FROM AUTHOR]

In this paper we focus on problems inapproximable in polynomial time and explore how quickly their approximability improves as the allowed running time is gradually increased from polynomial to (sub-)exponential. We tackle a number of problems: For Min Independent Dominating Set , Max Induced Path , Forest and Tree , for any r ( n ) , a simple, known scheme gives an approximation ratio of r in time roughly r n / r . We show that, if this running time could be significantly improved, the ETH would fail. For Max Minimal Vertex Cover we give a r -approximation in time 2 n / r . We match this with a similarly tight result. We also give a log ⁡ r -approximation for Min ATSP in time 2 n / r and an r -approximation for Max Grundy Coloring in time r n / r . Finally, we investigate the approximability of Min Set Cover , when measuring the running time as a function of the number of sets in the input. [ABSTRACT FROM AUTHOR]

Instance sparsification is well-known in the world of exact computation since it is very closely linked to the Exponential Time Hypothesis. In this paper, we extend the concept of sparsification in order to capture subexponential time approximation. We develop a new tool for inapproximability, called approximation preserving sparsification, and use it in order to get strong inapproximability results in subexponential time for several fundamental optimization problems such as min dominating set, min feedback vertex set, min set cover, min feedback arc set, and others. [ABSTRACT FROM AUTHOR]

The Graph Motif problem was introduced in 2006 in the context of biological networks. It consists of deciding whether or not a multiset of colors occurs in a connected subgraph of a vertex-colored graph. Graph Motif has been mostly analyzed from the standpoint of parameterized complexity. The main parameters which came into consideration were the size of the multiset and the number of colors. In the many utilizations of Graph Motif , however, the input graph originates from real-life applications and has structure. Motivated by this prosaic observation, we systematically study its complexity relatively to graph structural parameters. For a wide range of parameters, we give new or improved FPT algorithms, or show that the problem remains intractable. For the FPT cases, we also give some kernelization lower bounds as well as some ETH-based lower bounds on the worst case running time. Interestingly, we establish that Graph Motif is W[1] -hard (while in W[P] ) for parameter max leaf number, which is, to the best of our knowledge, the first problem to behave this way. [ABSTRACT FROM AUTHOR]