Edge Deletion Problem Research Articles

Dichotomy Results on the Hardness of $H$-free Edge Modification Problems

For a graph $H$, the $H$-free Edge Deletion problem asks whether there exist at most $k$ edges whose deletion from the input graph $G$ results in a graph without any induced copy of $H$. $H$-free Edge Completion and $H$-free Edge Editing are defined similarly where only completion (addition) of edges are allowed in the former and both completion and deletion are allowed in the latter. We completely settle the classical complexities of these problems by proving that $H$-free Edge Deletion is NP-complete if and only if $H$ is a graph with at least two edges, $H$-free Edge Completion is NP-complete if and only if $H$ is a graph with at least two nonedges, and $H$-free Edge Editing is NP-complete if and only if $H$ is a graph with at least three vertices. Our result on $H$-free Edge Editing resolves a conjecture by Alon and Stav [Theoret. Comput. Sci., 2009, pp. 4920--4927]. Additionally, we prove that these NP-complete problems cannot be solved in parameterized subexponential time, i.e., in time $2^{o(k)}\cd...

On Polynomial Kernelization of $$\mathcal {H}$$ H -free Edge Deletion

For a set $$\mathcal {H}$$ of graphs, the $$\mathcal {H}$$ -free Edge Deletion problem is to decide whether there exist at most k edges in the input graph, for some $$k\in \mathbb {N}$$ , whose deletion results in a graph without an induced copy of any of the graphs in $$\mathcal {H}$$ . The problem is known to be fixed-parameter tractable if $$\mathcal {H}$$ is of finite cardinality. In this paper, we present a polynomial kernel for this problem for any fixed finite set $$\mathcal {H}$$ of connected graphs for the case where the input graphs are of bounded degree. We use a single kernelization rule which deletes vertices ‘far away’ from the induced copies of every $$H\in \mathcal {H}$$ in the input graph. With a slightly modified kernelization rule, we obtain polynomial kernels for $$\mathcal {H}$$ -free Edge Deletion under the following three settings: where $$s>1$$ and $$t>2$$ are any fixed integers. Our result provides the first polynomial kernels for Claw-free Edge Deletion and Line Edge Deletion for $$K_t$$ -free input graphs which are known to be NP-complete even for $$K_4$$ -free graphs.

Polynomial Kernelization for Removing Induced Claws and Diamonds

A graph is called {claw,diamond}-free if it contains neither a claw (a K1,3) nor a diamond (a K4 with an edge removed) as an induced subgraph. Equivalently, {claw,diamond}-free graphs are characterized as line graphs of triangle-free graphs, or as linear dominoes (graphs in which every vertex is in at most two maximal cliques and every edge is in exactly one maximal clique). We consider the parameterized complexity of the {claw,diamond}-free Edge Deletion problem, where given a graph G and a parameter k, the question is whether one can remove at most k edges from G to obtain a {claw,diamond}-free graph. Our main result is that this problem admits a polynomial kernel. We complement this result by proving that, even on instances with maximum degree 6, the problem is NP-complete and cannot be solved in time 2^{o(k)}cdot |V(G)|^{mathcal {O}(1)} unless the Exponential Time Hypothesis fails.

Edge deletion problems: Branching facilitated by modular decomposition

Edge deletion problems ask for a minimum set of edges whose deletion makes a graph have a certain property. When this property can be characterized by a finite set of forbidden induced subgraphs, the problem can be solved in fixed-parameter time by a naive bounded search tree algorithm. Sometimes deleting an edge to break an erstwhile forbidden induced subgraph might introduce new ones, which may involve the neighbors of the original forbidden induced subgraph. Therefore, in considering possible ways to break a forbidden induced subgraph one naturally takes its neighborhood into consideration. This observation easily yields more efficient branching rules, but a naive implementation will require too many tedious case analyses. Here we take advantage of modular decomposition, which allows us to focus on far simpler quotient graphs instead of the original graphs. They together yield simple improved algorithms for the edge deletion problems to chain graphs and trivially perfect graphs.

Incompressibility of $$H$$ H -Free Edge Modification Problems

Given a fixed graph $$H$$H, the $$H$$H-Free Edge Deletion (resp., Completion, Editing) problem asks whether it is possible to delete from (resp., add to, delete from or add to) the input graph at most $$k$$k edges so that the resulting graph is $$H$$H-free, i.e., contains no induced subgraph isomorphic to $$H$$H. These $$H$$H-free edge modification problems are well known to be fixed-parameter tractable for every fixed $$H$$H. In this paper we study the incompressibility, i.e., nonexistence of polynomial kernels, for these $$H$$H-free edge modification problems in terms of the structure of $$H$$H, and completely characterize their nonexistence for $$H$$H being paths, cycles or 3-connected graphs. We also give a sufficient condition for the nonexistence of polynomial kernels for $${\mathcal {F}}$$F-Free Edge Deletion problems, where $${\mathcal {F}}$$F is a finite set of forbidden induced subgraphs. As an effective tool, we have introduced an incompressible constraint satisfiability problem Propagational-$$f$$f Satisfiability to express common propagational behaviors of events, and we expect the problem to be useful in studying the nonexistence of polynomial kernels in general.

An overview of kernelization algorithms for graph modification problems

Kernelization algorithms for graph modification problems are important ingredients in parameterized computation theory. In this paper, we survey the kernelization algorithms for four types of graph modification problems, which include vertex deletion problems, edge editing problems, edge deletion problems, and edge completion problems. For each type of problem, we outline typical examples together with recent results, analyze the main techniques, and provide some suggestions for future research in this field.

Two edge modification problems without polynomial kernels

Given a graph G and an integer k, the Π Edge Completion/Editing/Deletion problem asks whether it is possible to add, edit, or delete at most k edges in G such that one obtains a graph that fulfills the property Π. Edge modification problems have received considerable interest from a parameterized point of view. When parameterized by k, many of these problems turned out to be fixed-parameter tractable and some are known to admit polynomial kernelizations, i.e., efficient preprocessing with a size guarantee that is polynomial in k. This paper answers an open problem posed by Cai (IWPEC 2006) [11], namely, whether the Π Edge Deletion problem, parameterized by the number of deletions, admits a polynomial kernelization when Π can be characterized by a finite set of forbidden induced subgraphs. We answer this question negatively based on the framework for kernelization lower bounds of Bodlaender et al. (ICALP 2008, JCSS 2009) [15] and Fortnow and Santhanam (STOC 2008, JCSS 2011) [16]. We present a graph H on seven vertices such that H-free Edge Deletion and H-free Edge Editing do not admit polynomial kernelizations, unless NP⊆coNP/poly. The application of the framework is not immediate and requires a lower bound for a Not-1-in-3 SAT problem that may be of independent interest.

Contracting Graphs to Paths and Trees

Vertex deletion and edge deletion problems play a central role in parameterized complexity. Examples include classical problems like Feedback Vertex Set, Odd Cycle Transversal, and Chordal Deletion. The study of analogous edge contraction problems has so far been left largely unexplored from a parameterized perspective. We consider two basic problems of this type: Tree Contraction and Path Contraction. These two problems take as input an undirected graph G on n vertices and an integer k, and the task is to determine whether we can obtain a tree or a path, respectively, by a sequence of at most k edge contractions in G. For Tree Contraction, we present a randomized 4 k n O(1) time polynomial-space algorithm, as well as a deterministic 4.98 k n O(1) time algorithm, based on a variant of the color coding technique of Alon, Yuster and Zwick. We also present a deterministic 2 k+o(k)+n O(1) time algorithm for Path Contraction. Furthermore, we show that Path Contraction has a kernel with at most 5k+3 vertices, while Tree Contraction does not have a polynomial kernel unless NP ⊆ coNP/poly. We find the latter result surprising because of the connection between Tree Contraction and Feedback Vertex Set, which is known to have a kernel with 4k 2 vertices.

Parameterized Complexity of Eulerian Deletion Problems

We study a family of problems where the goal is to make a graph Eulerian, i.e., connected and with all the vertices having even degrees, by a minimum number of deletions. We completely classify the parameterized complexity of various versions: undirected or directed graphs, vertex or edge deletions, with or without the requirement of connectivity, etc. The collection of results shows an interesting contrast: while the node-deletion variants remain intractable, i.e., W[1]-hard for all the studied cases, edge-deletion problems are either fixed-parameter tractable or polynomial-time solvable. Of particular interest is a randomized FPT algorithm for making an undirected graph Eulerian by deleting the minimum number of edges, based on a novel application of the color coding technique. For versions that remain NP-complete but fixed-parameter tractable we consider also possibilities of polynomial kernelization; unfortunately, we prove that this is not possible unless NP⊆coNP/poly.

BOUNDED SEARCH TREE ALGORITHMS FOR PARAMETRIZED 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 P4-sparse graphs as the relaxed graph class, we obtain efficient bounded search tree algorithms for several parametrized 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 [R. Niedermeier and P. Rossmanith, An efficient fixed-parameter algorithm for 3-hitting set, J. Discrete Algorithms1(1) (2003) 89–102] and computer-aided branching rules [J. Gramm, J. Guo, F. Hüffner and R. Niedermeier, Automated generation of search tree algorithms for hard graph modification problems, Algorithmica39(4) (2004) 321–347].

On the (Non-)Existence of Polynomial Kernels for P l -Free Edge Modification Problems

Given a graph G=(V,E) and a positive integer k, an edge modification problem for a graph property ź consists in deciding whether there exists a set F of pairs of V of size at most k such that the graph $H=(V,E\vartriangle F)$ satisfies the property ź. In the źedge-completion problem, the set F is constrained to be disjoint from E; in the źedge-deletion problem, F is a subset of E; no constraint is imposed on F in the źedge-editing problem. A number of optimization problems can be expressed in terms of graph modification problems which have been extensively studied in the context of parameterized complexity (Cai in Inf. Process. Lett. 58:171---176, 1996; Fellows et al. in FCT, pp. 312---321, 2007; Heggernes et al. in STOC, pp. 374---381, 2007). When parameterized by the size k of the set F, it has been proved that if ź is a hereditary property characterized by a finite set of forbidden induced subgraphs, then the three ź edge-modification problems are FPT (Cai in Inf. Process. Lett. 58:171---176, 1996). It was then natural to ask (Bodlaender et al. in IWPEC, 2006) whether these problems also admit a polynomial kernel. in polynomial time to an equivalent instance (Gź,kź) with size bounded by a polynomial in k). Using recent lower bound techniques, Kratsch and Wahlstrom answered this question negatively (Kratsch and Wahlstrom in IWPEC, pp. 264---275, 2009). However, the problem remains open on many natural graph classes characterized by forbidden induced subgraphs. question to characterize for which type of graph properties, the parameterized edge-modification problems have polynomial kernels. Kratsch and Wahlstrom asked whether the result holds when the forbidden subgraphs are paths or cycles and pointed out that the problem is already open in the case of P4-free graphs (i.e. cographs). This paper provides positive and negative results in that line of research. We prove that Parameterized cograph edge-modification problems have cubic vertex kernels whereas polynomial kernels are unlikely to exist for the Pl-free edge-deletion and the Cl-free edge-deletion problems for lź7 and lź4 respectively. Indeed, if they exist, then NP⊆coNP/poly.

FPT algorithms for path-transversal and cycle-transversal problems

We study the parameterized complexity of several vertex- and edge-deletion problems on graphs, parameterized by the number p of deletions. The first kind of problems are separation problems on undirected graphs, where we aim at separating distinguished vertices in a graph. The second kind of problems are feedback set problems on group-labelled graphs, where we aim at breaking nonnull cycles in a graph having its arcs labelled by elements of a group. We obtain new FPT algorithms for these different problems, relying on a generic O ∗ ( 4 p ) algorithm for breaking paths of a homogeneous path system.

Additive approximation for edge-deletion problems

A graph property is monotone if it is closed under removal of vertices and edges. In this paper we consider the following algorithmic problem, called the edge-deletion problem; given a monotone property P and a graph G, compute the smallest number of edge deletions that are needed in order to turn G into a graph satisfying P. We denote this quantity by E' P (G). The first result of this paper states that the edge-deletion problem can be efficiently approximated for any monotone property. • For any fixed e > 0 and any monotone property P, there is a deterministic algorithm which, given a graph G = (V, E) of size n, approximates E' P (G) in linear time O(|V| + |E|) to within an additive error of en 2 . Given the above, a natural question is for which monotone properties one can obtain better additive approximations of E' P . Our second main result essentially resolves this problem by giving a precise characterization of the monotone graph properties for which such approximations exist. (1) If there is a bipartite graph that does not satisfy P, then there is a δ > 0 for which it is possible to approximate E' P to within an additive error of n 2 -δ in polynomial time. (2) On the other hand, if all bipartite graphs satisfy P, then for any δ > 0 it is NP-hard to approximate E' P to within an additive error of n 2-δ . While the proof of (1) is relatively simple, the proof of (2) requires several new ideas and involves tools from Extremal Graph Theory together with spectral techniques. Interestingly, prior to this work it was not even known that computing E' P precisely for the properties in (2) is NP-hard. We thus answer (in a strong form) a question of Yannakakis, who asked in 1981 if it is possible to find a large and natural family of graph properties for which computing E' P is NP-hard.

Hardness of edge-modification problems

For a graph property P consider the following computational problem. Given an input graph G , what is the minimum number of edge modifications (additions and/or deletions) that one has to apply to G in order to turn it into a graph that satisfies P ? Namely, what is the edit distance Δ ( G , P ) of a graph G from satisfying P ? Clearly, the computational complexity of such a problem strongly depends on P . For over 30 years this family of computational problems has been studied in several contexts and various algorithms, as well as hardness results, were obtained for specific graph properties. Alon, Shapira and Sudakov studied in [N. Alon, A. Shapira, B. Sudakov, Additive approximation for edge-deletion problems, in: Proc. of the 46th IEEE FOCS, 2005, 419–428. Also: Annals of Mathematics (in press)] the approximability of the computational problem for the family of monotone graph properties, namely properties that are closed under removal of edges and vertices. They describe an efficient algorithm that achieves an o ( n 2 ) additive approximation to Δ ( G , P ) for any monotone property P , where G is an n -vertex input graph, and show that the problem of achieving an O ( n 2 − ε ) additive approximation is N P -hard for most monotone properties. The methods in [N. Alon, A. Shapira, B. Sudakov, Additive approximation for edge-deletion problems, in: Proc. of the 46th IEEE FOCS, 2005, 419–428. Also: Annals of Mathematics (in press)] also provide a polynomial time approximation algorithm which computes Δ ( G , P ) ± o ( n 2 ) for the broader family of hereditary graph properties (which are closed under removal of vertices). In this work we introduce two approaches for showing that improving upon the additive approximation achieved by this algorithm is N P -hard for several sub-families of hereditary properties. In addition, we state a conjecture on the hardness of computing the edit distance from being induced H -free for any forbidden graph H .

Register Allocation in Structured Programs

In this article we look at the register allocation problem. In the literature this problem is frequently reduced to the general graph coloring problem and the solutions to the problem are obtained from graph coloring heuristics. Hence, no algorithm with a good performance guarantee is known. Here we show that when attention is restricted tostructured programswhich we define to be programs whose control-flow graphs are series-parallel, there is an efficient algorithm that produces a solution which is within a factor of 2 of the optimal solution. We note that even with the previous restriction the resulting coloring problem is NP-complete.We also consider how to delete a minimum number of edges from arbitrary control-flow graphs to make them series-parallel and to apply our algorithm. We show that this problem is MaxSNP hard. However, we define the notion of anapproximate articulation pointand we give efficient algorithms to find approximate articulation points. We present a heuristic for the edge deletion problem based on this notion which seems to work well when the given graph is close to series-parallel.

EDGE-DELETION GRAPH PROBLEMS WITH FIRST-ORDER EXPRESSIBLE SUBGRAPH PROPERTIES

In this paper edge-deletion problems are studied with a new perspective. In general an edge-deletion problem is of the form: Given a graph G, does it have a subgraph H obtained by deleting zero or more edges such that H satisfies a polynomial-time verifiable property? This paper restricts attention to first-order expressible properties. If the property is expressed by π, which in prenex normal form is Q(Φ) where Q is the quantifier-prefix, then we prove results on the quantifier structure that characterize the complexity of the edge-deletion problem. In particular we give polynomial-time algorithms for problems for which Q is ‘simple’ and in other cases we encode certain NP-complete problems as edge-deletion problems, essentially using the quantifier structure of π. We also present evidence that Q alone cannot capture the complexity of the edge-deletion problem.

Fair edge deletion problems

The notation of fair edge-deletion problems is introduced. These arise when it is desirable to control the number of edges incident to any node that are either deleted or remain following edge deletion. Six such problems were formulated for the case where the resultant graph is known to be acyclic, and the complexity of four of these is easily determined from known results. The remaining two are the authors' focus. It is shown that the problem of finding a minimum-degree deletion graph H such that G-H is acyclic is NP-hard when G is undirected, and is solvable in linear time when G is directed. >

The complexity of some edge deletion problems

The edge deletion problem (EDP) corresponding to a given class H of graphs is to find the minimum number of edges the deletion of which from a given graph G results in a subgraph G', G' in H. Previous complexity results are extended by showing that the EDP corresponding to any class H of graphs in each of the following cases is NP-hard. (1) H is defined by a set of forbidden homeomorphs or minors in which every member is a 2-connected graph with minimum degree three; (2) BH is defined by K/sub 4/-e as a forbidden homeomorph or minor; and (3) H is defined by P/sub l/, l>or=3, the simple path on l nodes, as a forbidden induced subgraph. >

An Application of Duality to Edge-Deletion Problems

For a property $\pi $ on graphs, the corresponding edge-deletion problem $P_{{\text{ED}}} (\pi )$ (on planar graphs) is stated as follows: given a (planar) graph G, find a set of edges of minimum cardinality whose deletion results in a graph satisfying $\pi $. We show that the edge-deletion problem $P_{{\text{ED}}} (\pi )$ on planar graphs is NP-hard if $\pi $ is nontrivial and is determined by the weighted 3-connected components. As a corollary, the edge-deletion problem $P_{{\text{ED}}} (\pi )$ on planar graphs is NP-complete for several properties $\pi $, such as $\pi = $ “series-parallel,” “outerplanar,” “without cocycles of cardinality $ \geqq 3$,” etc.

On the NP-hardness of edge-deletion and -contraction problems

If π is a property on graphs, the corresponding edge deletion (edge contraction, respectively) problem is: Given a graph G, determine the minimum number of edges of G whose deletion (contraction) results in a graph satisfying property π. We show that these problems are NP-hard if π is finitely characterizable by 3-connected graphs.