Hitting forbidden subgraphs in graphs of bounded treewidth

We study the complexity of a generic hitting problem H-Subgraph Hitting, where given a fixed pattern graph $H$ and an input graph $G$, the task is to find a set $X \subseteq V(G)$ of minimum size that hits all subgraphs of $G$ isomorphic to $H$. In the colorful variant of the problem, each vertex of $G$ is precolored with some color from $V(H)$ and we require to hit only $H$-subgraphs with matching colors. Standard techniques shows that for every fixed $H$, the problem is fixed-parameter tractable parameterized by the treewidth of $G$; however, it is not clear how exactly the running time should depend on treewidth. For the colorful variant, we demonstrate matching upper and lower bounds showing that the dependence of the running time on treewidth of $G$ is tightly governed by $\mu(H)$, the maximum size of a minimal vertex separator in $H$. That is, we show for every fixed $H$ that, on a graph of treewidth $t$, the colorful problem can be solved in time $2^{\mathcal{O}(t^{\mu(H)})}\cdot|V(G)|$, but cannot be solved in time $2^{o(t^{\mu(H)})}\cdot |V(G)|^{O(1)}$, assuming the Exponential Time Hypothesis (ETH). Furthermore, we give some preliminary results showing that, in the absence of colors, the parameterized complexity landscape of H-Subgraph Hitting is much richer.Comment: A full version of a paper presented at MFCS 201

Hitting forbidden subgraphs in graphs of bounded treewidth

We study the complexity of a generic hitting problem H -Subgraph Hitting , where given a fixed pattern graph H and an input graph G, we seek for the minimum size of a set X ⊆ V(G) that hits all subgraphs of G isomorphic to H. In the colorful variant of the problem, each vertex of G is precolored with some color from V(H) and we require to hit only H-subgraphs with matching colors. Standard techniques (e.g., Courcelle’s theorem) show that, for every fixed H and the problem is fixed-parameter tractable parameterized by the treewidth of G; however, it is not clear how exactly the running time should depend on treewidth. For the colorful variant, we demonstrate matching upper and lower bounds showing that the dependence of the running time on treewidth of G is tightly governed by μ(H), the maximum size of a minimal vertex separator in H. That is, we show for every fixed H that, on a graph of treewidth t, the colorful problem can be solved in time 2O(tμ(H))⋅|V(G)|, but cannot be solved in time 2o(tμ(H))⋅|V(G)|O(1), assuming the Exponential Time Hypothesis (ETH). Furthermore, we give some preliminary results showing that, in the absence of colors, the parameterized complexity landscape of H -Subgraph Hitting is much richer

Hitting forbidden minors: Approximation and Kernelization

We study a general class of problems called F-deletion problems. In an F-deletion problem, we are asked whether a subset of at most $k$ vertices can be deleted from a graph $G$ such that the resulting graph does not contain as a minor any graph from the family F of forbidden minors. We obtain a number of algorithmic results on the F-deletion problem when F contains a planar graph. We give (1) a linear vertex kernel on graphs excluding $t$-claw $K_{1,t}$, the star with $t$ leves, as an induced subgraph, where $t$ is a fixed integer. (2) an approximation algorithm achieving an approximation ratio of $O(\log^{3/2} OPT)$, where $OPT$ is the size of an optimal solution on general undirected graphs. Finally, we obtain polynomial kernels for the case when F contains graph $\theta_c$ as a minor for a fixed integer $c$. The graph $\theta_c$ consists of two vertices connected by $c$ parallel edges. Even though this may appear to be a very restricted class of problems it already encompasses well-studied problems such as {\sc Vertex Cover}, {\sc Feedback Vertex Set} and Diamond Hitting Set. The generic kernelization algorithm is based on a non-trivial application of protrusion techniques, previously used only for problems on topological graph classes

Bounded Search Tree Algorithms for Parameterized Cograph Deletion: Efficient Branching Rules by Exploiting Structures of Special Graph Classes

Many fixed-parameter tractable algorithms using a bounded search tree have been repeatedly improved, often by describing a larger number of branching rules involving an increasingly complex case analysis. We introduce a novel and general search strategy that branches on the forbidden subgraphs of a graph class relaxation. By using the class of $P_4$-sparse graphs as the relaxed graph class, we obtain efficient bounded search tree algorithms for several parameterized deletion problems. We give the first non-trivial bounded search tree algorithms for the cograph edge-deletion problem and the trivially perfect edge-deletion problems. For the cograph vertex deletion problem, a refined analysis of the runtime of our simple bounded search algorithm gives a faster exponential factor than those algorithms designed with the help of complicated case distinctions and non-trivial running time analysis [21] and computer-aided branching rules [11].Comment: 23 pages. Accepted in Discrete Mathematics, Algorithms and Applications (DMAA

Meta-Kernelization using Well-Structured Modulators

Kernelization investigates exact preprocessing algorithms with performance guarantees. The most prevalent type of parameters used in kernelization is the solution size for optimization problems; however, also structural parameters have been successfully used to obtain polynomial kernels for a wide range of problems. Many of these parameters can be defined as the size of a smallest modulator of the given graph into a fixed graph class (i.e., a set of vertices whose deletion puts the graph into the graph class). Such parameters admit the construction of polynomial kernels even when the solution size is large or not applicable. This work follows up on the research on meta-kernelization frameworks in terms of structural parameters. We develop a class of parameters which are based on a more general view on modulators: instead of size, the parameters employ a combination of rank-width and split decompositions to measure structure inside the modulator. This allows us to lift kernelization results from modulator-size to more general parameters, hence providing smaller kernels. We show (i) how such large but well-structured modulators can be efficiently approximated, (ii) how they can be used to obtain polynomial kernels for any graph problem expressible in Monadic Second Order logic, and (iii) how they allow the extension of previous results in the area of structural meta-kernelization

Hitting forbidden induced subgraphs on bounded treewidth graphs

For a fixed graph $H$, the $H$-IS-Deletion problem asks, given a graph $G$, for the minimum size of a set $S \subseteq V(G)$ such that $G\setminus S$ does not contain $H$ as an induced subgraph. Motivated by previous work about hitting (topological) minors and subgraphs on bounded treewidth graphs, we are interested in determining, for a fixed graph $H$, the smallest function $f_H(t)$ such that $H$-IS-Deletion can be solved in time $f_H(t) \cdot n^{O(1)}$ assuming the Exponential Time Hypothesis (ETH), where $t$ and $n$ denote the treewidth and the number of vertices of the input graph, respectively. We show that $f_H(t) = 2^{O(t^{h-2})}$ for every graph $H$ on $h \geq 3$ vertices, and that $f_H(t) = 2^{O(t)}$ if $H$ is a clique or an independent set. We present a number of lower bounds by generalizing a reduction of Cygan et al. [MFCS 2014] for the subgraph version. In particular, we show that when $H$ deviates slightly from a clique, the function $f_H(t)$ suffers a sharp jump: if $H$ is obtained from a clique of size $h$ by removing one edge, then $f_H(t) = 2^{\Theta(t^{h-2})}$. We also show that $f_H(t) = 2^{\Omega(t^{h})}$ when $H=K_{h,h}$, and this reduction answers an open question of Mi. Pilipczuk [MFCS 2011] about the function $f_{C_4}(t)$ for the subgraph version. Motivated by Cygan et al. [MFCS 2014], we also consider the colorful variant of the problem, where each vertex of $G$ is colored with some color from $V(H)$ and we require to hit only induced copies of $H$ with matching colors. In this case, we determine, under the ETH, the function $f_H(t)$ for every connected graph $H$ on $h$ vertices: if $h\leq 2$ the problem can be solved in polynomial time; if $h\geq 3$, $f_H(t) = 2^{\Theta(t)}$ if $H$ is a clique, and $f_H(t) = 2^{\Theta(t^{h-2})}$ otherwise.Comment: 24 pages, 3 figure

Hitting Forbidden Induced Subgraphs on Bounded Treewidth Graphs

For a fixed graph H, the H-IS-Deletion problem asks, given a graph G, for the minimum size of a set S ? V(G) such that G? S does not contain H as an induced subgraph. Motivated by previous work about hitting (topological) minors and subgraphs on bounded treewidth graphs, we are interested in determining, for a fixed graph H, the smallest function f_H(t) such that H-IS-Deletion can be solved in time f_H(t) ? n^{?(1)} assuming the Exponential Time Hypothesis (ETH), where t and n denote the treewidth and the number of vertices of the input graph, respectively. We show that f_H(t) = 2^{?(t^{h-2})} for every graph H on h ? 3 vertices, and that f_H(t) = 2^{?(t)} if H is a clique or an independent set. We present a number of lower bounds by generalizing a reduction of Cygan et al. [MFCS 2014] for the subgraph version. In particular, we show that when H deviates slightly from a clique, the function f_H(t) suffers a sharp jump: if H is obtained from a clique of size h by removing one edge, then f_H(t) = 2^{?(t^{h-2})}. We also show that f_H(t) = 2^{?(t^{h})} when H = K_{h,h}, and this reduction answers an open question of Mi. Pilipczuk [MFCS 2011] about the function f_{C?}(t) for the subgraph version. Motivated by Cygan et al. [MFCS 2014], we also consider the colorful variant of the problem, where each vertex of G is colored with some color from V(H) and we require to hit only induced copies of H with matching colors. In this case, we determine, under the ETH, the function f_H(t) for every connected graph H on h vertices: if h ? 2 the problem can be solved in polynomial time; if h ? 3, f_H(t) = 2^{?(t)} if H is a clique, and f_H(t) = 2^{?(t^{h-2})} otherwise

Hitting Subgraphs in Sparse Graphs and Geometric Intersection Graphs

We investigate a fundamental vertex-deletion problem called (Induced) Subgraph Hitting: given a graph $G$ and a set $\mathcal{F}$ of forbidden graphs, the goal is to compute a minimum-sized set $S$ of vertices of $G$ such that $G-S$ does not contain any graph in $\mathcal{F}$ as an (induced) subgraph. This is a generic problem that encompasses many well-known problems that were extensively studied on their own, particularly (but not only) from the perspectives of both approximation and parameterization. We focus on the design of efficient approximation schemes, i.e., with running time $f(\varepsilon,\mathcal{F}) \cdot n^{O(1)}$, which are also of significant interest to both communities. Technically, our main contribution is a linear-time approximation-preserving reduction from (Induced) Subgraph Hitting on any graph class $\mathcal{G}$ of bounded expansion to the same problem on bounded degree graphs within $\mathcal{G}$. This yields a novel algorithmic technique to design (efficient) approximation schemes for the problem on very broad graph classes, well beyond the state-of-the-art. Specifically, applying this reduction, we derive approximation schemes with (almost) linear running time for the problem on any graph classes that have strongly sublinear separators and many important classes of geometric intersection graphs (such as fat-object graphs, pseudo-disk graphs, etc.). Our proofs introduce novel concepts and combinatorial observations that may be of independent interest (and, which we believe, will find other uses) for studies of approximation algorithms, parameterized complexity, sparse graph classes, and geometric intersection graphs. As a byproduct, we also obtain the first robust algorithm for $k$-Subgraph Isomorphism on intersection graphs of fat objects and pseudo-disks, with running time $f(k) \cdot n \log n + O(m)$.Comment: 60 pages, abstract shortened to fulfill the length limi