186 research outputs found

    Parameterized Complexity of Deletion to Scattered Graph Classes

    Get PDF
    Graph-modification problems, where we add/delete a small number of vertices/edges to make the given graph to belong to a simpler graph class, is a well-studied optimization problem in all algorithmic paradigms including classical, approximation and parameterized complexity. Specifically, graph-deletion problems, where one needs to delete at most k vertices to place it in a given non-trivial hereditary (closed under induced subgraphs) graph class, captures several well-studied problems including Vertex Cover, Feedback Vertex Set, Odd Cycle Transveral, Cluster Vertex Deletion, and Perfect Deletion. Investigation into these problems in parameterized complexity has given rise to powerful tools and techniques. While a precise characterization of the graph classes for which the problem is fixed-parameter tractable (FPT) is elusive, it has long been known that if the graph class is characterized by a finite set of forbidden graphs, then the problem is FPT. In this paper, we initiate a study of a natural variation of the problem of deletion to scattered graph classes where we need to delete at most k vertices so that in the resulting graph, each connected component belongs to one of a constant number of graph classes. A simple hitting set based approach is no longer feasible even if each of the graph classes is characterized by finite forbidden sets. As our main result, we show that this problem (in the case where each graph class has a finite forbidden set) is fixed-parameter tractable by a O^*(2^(k^O(1))) algorithm, using a combination of the well-known techniques in parameterized complexity - iterative compression and important separators. Our approach follows closely that of a related problem in the context of satisfiability [Ganian, Ramanujan, Szeider, TAlg 2017], where one wants to find a small backdoor set so that the resulting CSP (constraint satisfaction problem) instance belongs to one of several easy instances of satisfiability. While we follow the main idea from this work, there are some challenges for our problem which we needed to overcome. When there are two graph classes with finite forbidden sets to get to, and if one of the forbidden sets has a path, then we show that the problem has a (better) singly exponential algorithm and a polynomial sized kernel. We also design an efficient FPT algorithm for a special case when one of the graph classes has an infinite forbidden set. Specifically, we give a O^*(4^k) algorithm to determine whether k vertices can be deleted from a given graph so that in the resulting graph, each connected component is a tree (the sparsest connected graph) or a clique (the densest connected graph)

    Reoptimization of Some Maximum Weight Induced Hereditary Subgraph Problems

    Get PDF
    The reoptimization issue studied in this paper can be described as follows: given an instance I of some problem Π, an optimal solution OPT for Π in I and an instance I′ resulting from a local perturbation of I that consists of insertions or removals of a small number of data, we wish to use OPT in order to solve Π in I', either optimally or by guaranteeing an approximation ratio better than that guaranteed by an ex nihilo computation and with running time better than that needed for such a computation. We use this setting in order to study weighted versions of several representatives of a broad class of problems known in the literature as maximum induced hereditary subgraph problems. The main problems studied are max independent set, max k-colorable subgraph and max split subgraph under vertex insertions and deletion

    Hitting forbidden minors: Approximation and Kernelization

    Get PDF
    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 kk vertices can be deleted from a graph GG 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 tt-claw K1,tK_{1,t}, the star with tt leves, as an induced subgraph, where tt is a fixed integer. (2) an approximation algorithm achieving an approximation ratio of O(log3/2OPT)O(\log^{3/2} OPT), where OPTOPT is the size of an optimal solution on general undirected graphs. Finally, we obtain polynomial kernels for the case when F contains graph θc\theta_c as a minor for a fixed integer cc. The graph θc\theta_c consists of two vertices connected by cc 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

    Polynomial kernels for 3-leaf power graph modification problems

    Full text link
    A graph G=(V,E) is a 3-leaf power iff there exists a tree T whose leaves are V and such that (u,v) is an edge iff u and v are at distance at most 3 in T. The 3-leaf power graph edge modification problems, i.e. edition (also known as the closest 3-leaf power), completion and edge-deletion, are FTP when parameterized by the size of the edge set modification. However polynomial kernel was known for none of these three problems. For each of them, we provide cubic kernels that can be computed in linear time for each of these problems. We thereby answer an open problem first mentioned by Dom, Guo, Huffner and Niedermeier (2005).Comment: Submitte

    Single-Exponential FPT Algorithms for Enumerating Secluded F\mathcal{F}-Free Subgraphs and Deleting to Scattered Graph Classes

    Full text link
    The celebrated notion of important separators bounds the number of small (S,T)(S,T)-separators in a graph which are 'farthest from SS' in a technical sense. In this paper, we introduce a generalization of this powerful algorithmic primitive that is phrased in terms of kk-secluded vertex sets: sets with an open neighborhood of size at most kk. In this terminology, the bound on important separators says that there are at most 4k4^k maximal kk-secluded connected vertex sets CC containing SS but disjoint from TT. We generalize this statement significantly: even when we demand that G[C]G[C] avoids a finite set F\mathcal{F} of forbidden induced subgraphs, the number of such maximal subgraphs is 2O(k)2^{O(k)} and they can be enumerated efficiently. This allows us to make significant improvements for two problems from the literature. Our first application concerns the 'Connected kk-Secluded F\mathcal{F}-free subgraph' problem, where F\mathcal{F} is a finite set of forbidden induced subgraphs. Given a graph in which each vertex has a positive integer weight, the problem asks to find a maximum-weight connected kk-secluded vertex set CV(G)C \subseteq V(G) such that G[C]G[C] does not contain an induced subgraph isomorphic to any FFF \in \mathcal{F}. The parameterization by kk is known to be solvable in triple-exponential time via the technique of recursive understanding, which we improve to single-exponential. Our second application concerns the deletion problem to scattered graph classes. Here, the task is to find a vertex set of size at most kk whose removal yields a graph whose each connected component belongs to one of the prescribed graph classes Π1,,Πd\Pi_1, \ldots, \Pi_d. We obtain a single-exponential algorithm whenever each class Πi\Pi_i is characterized by a finite number of forbidden induced subgraphs. This generalizes and improves upon earlier results in the literature.Comment: To appear at ISAAC'2

    Assessing the Computational Complexity of Multi-Layer Subgraph Detection

    Get PDF
    Multi-layer graphs consist of several graphs (layers) over the same vertex set. They are motivated by real-world problems where entities (vertices) are associated via multiple types of relationships (edges in different layers). We chart the border of computational (in)tractability for the class of subgraph detection problems on multi-layer graphs, including fundamental problems such as maximum matching, finding certain clique relaxations (motivated by community detection), or path problems. Mostly encountering hardness results, sometimes even for two or three layers, we can also spot some islands of tractability

    Tight Bounds for Chordal/Interval Vertex Deletion Parameterized by Treewidth

    Get PDF
    corecore