On Directed Feedback Vertex Set parameterized by treewidth

We study the Directed Feedback Vertex Set problem parameterized by the treewidth of the input graph. We prove that unless the Exponential Time Hypothesis fails, the problem cannot be solved in time $2^{o(t\log t)}\cdot n^{\mathcal{O}(1)}$ on general directed graphs, where $t$ is the treewidth of the underlying undirected graph. This is matched by a dynamic programming algorithm with running time $2^{\mathcal{O}(t\log t)}\cdot n^{\mathcal{O}(1)}$. On the other hand, we show that if the input digraph is planar, then the running time can be improved to $2^{\mathcal{O}(t)}\cdot n^{\mathcal{O}(1)}$.Comment: 20

On Directed Feedback Vertex Set parameterized by treewidth

We study the Directed Feedback Vertex Set problem parameterized by the treewidth of the input graph. We prove that unless the Exponential Time Hypothesis fails, the problem cannot be solved in time $2^{o(t\log t)}\cdot n^{\mathcal{O}(1)}$ on general directed graphs, where $t$ is the treewidth of the underlying undirected graph. This is matched by a dynamic programming algorithm with running time $2^{\mathcal{O}(t\log t)}\cdot n^{\mathcal{O}(1)}$. On the other hand, we show that if the input digraph is planar, then the running time can be improved to $2^{\mathcal{O}(t)}\cdot n^{\mathcal{O}(1)}$

Beyond Bidimensionality: Parameterized Subexponential Algorithms on Directed Graphs

We develop two different methods to achieve subexponential time parameterized algorithms for problems on sparse directed graphs. We exemplify our approaches with two well studied problems. For the first problem, {\sc $k$-Leaf Out-Branching}, which is to find an oriented spanning tree with at least $k$ leaves, we obtain an algorithm solving the problem in time $2^{O(\sqrt{k} \log k)} n+ n^{O(1)}$ on directed graphs whose underlying undirected graph excludes some fixed graph $H$ as a minor. For the special case when the input directed graph is planar, the running time can be improved to $2^{O(\sqrt{k})}n + n^{O(1)}$. The second example is a generalization of the {\sc Directed Hamiltonian Path} problem, namely {\sc $k$-Internal Out-Branching}, which is to find an oriented spanning tree with at least $k$ internal vertices. We obtain an algorithm solving the problem in time $2^{O(\sqrt{k} \log k)} + n^{O(1)}$ on directed graphs whose underlying undirected graph excludes some fixed apex graph $H$ as a minor. Finally, we observe that for any $\epsilon>0$, the {\sc $k$-Directed Path} problem is solvable in time $O((1+\epsilon)^k n^{f(\epsilon)})$, where $f$ is some function of \ve. Our methods are based on non-trivial combinations of obstruction theorems for undirected graphs, kernelization, problem specific combinatorial structures and a layering technique similar to the one employed by Baker to obtain PTAS for planar graphs

Parameterization Above a Multiplicative Guarantee

Parameterization above a guarantee is a successful paradigm in Parameterized Complexity. To the best of our knowledge, all fixed-parameter tractable problems in this paradigm share an additive form defined as follows. Given an instance (I,k) of some (parameterized) problem ? with a guarantee g(I), decide whether I admits a solution of size at least (at most) k+g(I). Here, g(I) is usually a lower bound (resp. upper bound) on the maximum (resp. minimum) size of a solution. Since its introduction in 1999 for Max SAT and Max Cut (with g(I) being half the number of clauses and half the number of edges, respectively, in the input), analysis of parameterization above a guarantee has become a very active and fruitful topic of research. We highlight a multiplicative form of parameterization above a guarantee: Given an instance (I,k) of some (parameterized) problem ? with a guarantee g(I), decide whether I admits a solution of size at least (resp. at most) k ? g(I). In particular, we study the Long Cycle problem with a multiplicative parameterization above the girth g(I) of the input graph, and provide a parameterized algorithm for this problem. Apart from being of independent interest, this exemplifies how parameterization above a multiplicative guarantee can arise naturally. We also show that, for any fixed constant ?>0, multiplicative parameterization above g(I)^(1+?) of Long Cycle yields para-NP-hardness, thus our parameterization is tight in this sense. We complement our main result with the design (or refutation of the existence) of algorithms for other problems parameterized multiplicatively above girth

Finding Long Directed Cycles Is Hard Even When DFVS Is Small or Girth Is Large

We study the parameterized complexity of two classic problems on directed graphs: Hamiltonian Cycle and its generalization Longest Cycle. Since 2008, it is known that Hamiltonian Cycle is W[1]-hard when parameterized by directed treewidth [Lampis et al., ISSAC\u2708]. By now, the question of whether it is FPT parameterized by the directed feedback vertex set (DFVS) number has become a longstanding open problem. In particular, the DFVS number is the largest natural directed width measure studied in the literature. In this paper, we provide a negative answer to the question, showing that even for the DFVS number, the problem remains W[1]-hard. As a consequence, we also obtain that Longest Cycle is W[1]-hard on directed graphs when parameterized multiplicatively above girth, in contrast to the undirected case. This resolves an open question posed by Fomin et al. [ACM ToCT\u2721] and Gutin and Mnich [arXiv:2207.12278]. Our hardness results apply to the path versions of the problems as well. On the positive side, we show that Longest Path parameterized multiplicatively above girth belongs to the class XP

Parameterized Complexity of Graph Constraint Logic

Graph constraint logic is a framework introduced by Hearn and Demaine, which provides several problems that are often a convenient starting point for reductions. We study the parameterized complexity of Constraint Graph Satisfiability and both bounded and unbounded versions of Nondeterministic Constraint Logic (NCL) with respect to solution length, treewidth and maximum degree of the underlying constraint graph as parameters. As a main result we show that restricted NCL remains PSPACE-complete on graphs of bounded bandwidth, strengthening Hearn and Demaine's framework. This allows us to improve upon existing results obtained by reduction from NCL. We show that reconfiguration versions of several classical graph problems (including independent set, feedback vertex set and dominating set) are PSPACE-complete on planar graphs of bounded bandwidth and that Rush Hour, generalized to $k\times n$ boards, is PSPACE-complete even when $k$ is at most a constant

Covering Small Independent Sets and Separators with Applications to Parameterized Algorithms

We present two new combinatorial tools for the design of parameterized algorithms. The first is a simple linear time randomized algorithm that given as input a $d$-degenerate graph $G$ and an integer $k$, outputs an independent set $Y$, such that for every independent set $X$ in $G$ of size at most $k$, the probability that $X$ is a subset of $Y$ is at least $\left({(d+1)k \choose k} \cdot k(d+1)\right)^{-1}$.The second is a new (deterministic) polynomial time graph sparsification procedure that given a graph $G$, a set $T = \{\{s_1, t_1\}, \{s_2, t_2\}, \ldots, \{s_\ell, t_\ell\}\}$ of terminal pairs and an integer $k$, returns an induced subgraph $G^\star$ of $G$ that maintains all the inclusion minimal multicuts of $G$ of size at most $k$, and does not contain any $(k+2)$-vertex connected set of size $2^{{\cal O}(k)}$. In particular, $G^\star$ excludes a clique of size $2^{{\cal O}(k)}$ as a topological minor. Put together, our new tools yield new randomized fixed parameter tractable (FPT) algorithms for Stable $s$-$t$ Separator, Stable Odd Cycle Transversal and Stable Multicut on general graphs, and for Stable Directed Feedback Vertex Set on $d$-degenerate graphs, resolving two problems left open by Marx et al. [ACM Transactions on Algorithms, 2013]. All of our algorithms can be derandomized at the cost of a small overhead in the running time.Comment: 35 page

Width Parameterizations for Knot-Free Vertex Deletion on Digraphs

A knot in a directed graph G is a strongly connected subgraph Q of G with at least two vertices, such that no vertex in V(Q) is an in-neighbor of a vertex in V(G)V(Q). Knots are important graph structures, because they characterize the existence of deadlocks in a classical distributed computation model, the so-called OR-model. Deadlock detection is correlated with the recognition of knot-free graphs as well as deadlock resolution is closely related to the Knot-Free Vertex Deletion (KFVD) problem, which consists of determining whether an input graph G has a subset S subseteq V(G) of size at most k such that G[VS] contains no knot. Because of natural applications in deadlock resolution, KFVD is closely related to Directed Feedback Vertex Set. In this paper we focus on graph width measure parameterizations for KFVD. First, we show that: (i) KFVD parameterized by the size of the solution k is W[1]-hard even when p, the length of a longest directed path of the input graph, as well as kappa, its Kenny-width, are bounded by constants, and we remark that KFVD is para-NP-hard even considering many directed width measures as parameters, but in FPT when parameterized by clique-width; (ii) KFVD can be solved in time 2^{O(tw)} x n, but assuming ETH it cannot be solved in 2^{o(tw)} x n^{O(1)}, where tw is the treewidth of the underlying undirected graph. Finally, since the size of a minimum directed feedback vertex set (dfv) is an upper bound for the size of a minimum knot-free vertex deletion set, we investigate parameterization by dfv and we show that (iii) KFVD can be solved in FPT-time parameterized by either dfv+kappa or dfv+p. Results of (iii) cannot be improved when replacing dfv by k due to (i)