Deletion Problem Research Articles

An approximation algorithm for the l-pseudoforest deletion problem

An l-pseudoforest is a graph each of whose connected components is at most l edges removal being a tree. The l-Pseudoforest Deletion problem is to delete a vertex set P of minimum weight from a given vertex-weighted graph G=(V,E) such that the remaining graph G[V∖P] is an l-pseudoforest. The Feedback Vertex Set problem is a special case of the l-Pseudoforest Deletion problem with l=0. In this paper, we present a polynomial time 4l-approximation algorithm for the l-Pseudoforest Deletion problem with l≥1 by using the local ratio technique. When l=1, we get a better approximation ratio 2 for the problem by further analyzing the local ratio, which matches the current best constant approximation factor for the Feedback Vertex Set problem.

Parameterized complexity of fair deletion problems

Edge deletion problems are those where the goal is to find a subset of edges such that after its removal the graph satisfies the given graph property. Typically, we want to minimize the number of elements removed. In fair deletion problems, the objective is changed, so the maximum number of deletions in a neighborhood of a single vertex is minimized.We study the parameterized complexity of fair deletion problems concerning the structural parameters such as the tree-width, the path-width, the tree-depth, the size of minimum feedback vertex set, the neighborhood diversity, and the size of minimum vertex cover of graph G.We prove the W[1]-hardness of the fair FO vertex-deletion problem with respect to the combined size of the tree-depth and the minimum feedback vertex set number. Moreover, we show that there is no algorithm for fair FO vertex-deletion problem running in time no(k3), where n is the size of the graph and k is the sum of the mentioned parameters, provided that the Exponential Time Hypothesis holds.On the other hand, we present an FPT algorithm for the fair MSO edge-deletion problem parameterized by the size of minimum vertex cover and an FPT algorithm for the fair MSO vertex-deletion problem parameterized by the neighborhood diversity.

Generalized Feedback Vertex Set Problems on Bounded-Treewidth Graphs: Chordality is the Key to Single-Exponential Parameterized Algorithms

It has long been known that Feedback Vertex Set can be solved in time $2^{\mathcal{O}(w\log w)}n^{\mathcal{O}(1)}$ on $n$-vertex graphs of treewidth $w$, but it was only recently that this running time was improved to $2^{\mathcal{O}(w)}n^{\mathcal{O}(1)}$, 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 $\mathcal{P}$ of graphs, the Bounded $\mathcal{P}$-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 $\mathcal{P}$. Assuming that $\mathcal{P}$ 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 $\mathcal{P}$ consists only of chordal graphs, then the problem can be solved in time $2^{\mathcal{O}(wd^2)} n^{\mathcal{O}(1)}$, and if $\mathcal{P}$ contains a graph with an induced cycle of length $\ell\ge 4$, then the problem is not solvable in time $2^{o(w\log w)} n^{\mathcal{O}(1)}$ even for fixed $d=\ell$, unless the ETH fails. We also study a similar problem, called Bounded $\mathcal{P}$-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 $\mathcal{P}$ contains all chordal graphs, then it cannot be solved in time $f(w)n^{o(w)}$ for some function $f$, unless the ETH fails.

Triangle edge deletion on planar glasses-free RGB-digraphs

In this paper, we revisit the triangle edge deletion problem a variant of feedback arc set problem which is one of the Karp's 21 NP-complete problems. Given a graph G(V,E), it is to find the edge set F such that |F|≤k and G′(V,E∖F) is a triangle-free graph. Yannakakis [1] showed it is hard for simple digraphs as well. The latest results showed that triangle edge deletion problem is still hard even for (a) planar undirected graphs of maximum degree seven [2] and (b) RGB-digraphs [3] which is a special class of digraphs. To make a further step, we show a new result in this paper that triangle edge deletion problem is still NP-complete even for planar RGB-digraphs of maximum degree eight with no glasses. On the other hand, we can find a kernel consisting of 11k/3 vertices for planar RGB-digraphs with no glasses. The results have an interesting application for the derivation of a tighter dichotomy on resilience decision problem in database theory.

A single-exponential fixed-parameter algorithm for distance-hereditary vertex deletion

Vertex deletion problems ask whether it is possible to delete at most k vertices from a graph so that the resulting graph belongs to a specified graph class. Over the past years, the parameterized complexity of vertex deletion to a plethora of graph classes has been systematically researched. Here we present the first single-exponential fixed-parameter tractable algorithm for vertex deletion to distance-hereditary graphs, a well-studied graph class which is particularly important in the context of vertex deletion due to its connection to the graph parameter rank-width. We complement our result with matching asymptotic lower bounds based on the exponential time hypothesis. As an application of our algorithm, we show that a vertex deletion set to distance-hereditary graphs can be used as a parameter which allows single-exponential fixed-parameter tractable algorithms for classical NP-hard problems.

Vertex deletion problems on chordal graphs

Containing many classic optimization problems, the family of vertex deletion problems has an important position in algorithm and complexity study. The celebrated result of Lewis and Yannakakis gives a complete dichotomy of their complexity. It however has nothing to say about the case when the input graph is also special. This paper initiates a systematic study of vertex deletion problems from one subclass of chordal graphs to another. We give polynomial-time algorithms or proofs of NP-completeness for most of the problems. In particular, we show that the vertex deletion problem from chordal graphs to interval graphs is NP-complete.

Strong triadic closure in cographs and graphs of low maximum degree

The MaxSTC problem is an assignment of the edges with two types of labels, namely, strong and weak, that maximizes the number of strong edges such that any two vertices that have a common neighbor with a strong edge are adjacent. The Cluster Deletion problem seeks for the minimum number of edge removals of a given graph such that the remaining graph is a disjoint union of cliques. Both problems are known to be NP-hard and an optimal solution for the Cluster Deletion problem provides a feasible solution for the MaxSTC problem, however not necessarily an optimal one. In this work we conduct the first systematic study that reveals graph families for which the optimal solutions for MaxSTC and Cluster Deletion coincide. We first show that MaxSTC coincides with Cluster Deletion on cographs and, thus, MaxSTC is solvable in polynomial time on cographs. As a side result, we give an interesting computational characterization of the maximum independent set on the cartesian product of two cographs. Furthermore, we address the influence of the low degree bounds to the complexity of the MaxSTC problem. We show that this problem is polynomial-time solvable on graphs of maximum degree three, whereas MaxSTC becomes NP-complete on graphs of maximum degree four. The proof of the latter result implies that there is no subexponential-time algorithm for MaxSTC unless the Exponential-Time Hypothesis fails.

Complexity of Optimally Defending and Attacking a Network

We consider single-agent, two-agent turn-taking and cooperative security games with Inverse Geodesic Length (IGL) as utility metric. We focus on the single-agent vertex deletion problem corresponding to IGL called MinIGL. Specifically, given a graph G, a budget k and a target inverse geodesic length T, does there exist a subset of vertices S of size k such that by deleting S the graph induced on the remaining vertices in G has IGL at most T. We cite our recently published work to report the results on the computational and parameterized complexity of MinIGL. Furthermore, we briefly state the problems we are interested to study in future.

Finding even subgraphs even faster

In the Undirected Eulerian Edge Deletion problem, we are given an undirected graph and an integer k, and the objective is to delete k edges such that the resultant graph is a connected graph in which all the vertices have even degrees. The corresponding problem in digraphs where the resulting graph should be strongly connected and every vertex should have the same in-degree as its out-degree is called Directed Eulerian Edge Deletion. In this paper, using the technique of computing representative families of co-graphic matroids we design algorithms which solve these problems in time 2O(k)nO(1), improving the algorithms by Cygan et al. [Algorithmica, 2014] and affirmatively answer the open problem posed by them. The crucial insight we bring to these problems is to view the solution as an independent set of a co-graphic matroid.

Tight Running Time Lower Bounds for Vertex Deletion Problems

For a graph class Π, the Π-V ertex D eletion problem has as input an undirected graph G = ( V , E ) and an integer k and asks whether there is a set of at most k vertices that can be deleted from G such that the resulting graph is a member of Π. By a classic result of Lewis and Yannakakis [17], Π-V ertex D eletion is NP-hard for all hereditary properties Π. We adapt the original NP-hardness construction to show that under the exponential time hypothesis (ETH), tight complexity results can be obtained. We show that Π-V ertex D eletion does not admit a 2 o ( n ) -time algorithm where n is the number of vertices in G . We also obtain a dichotomy for running time bounds that include the number m of edges in the input graph. On the one hand, if Π contains all edgeless graphs, then there is no 2 o ( n+m ) -time algorithm for Π-V ertex D eletion . On the other hand, if there is a fixed edgeless graph that is not contained in Π and containment in Π can be determined in 2 O ( n ) time or 2 o ( m ) time, then Π-V ertex D eletion can be solved in 2 O (√ m ) + O ( n ) or 2 o ( m ) + O ( n ) time, respectively. We also consider restrictions on the domain of the input graph G . For example, we obtain that Π-V ertex D eletion cannot be solved in 2 o (√ n ) time if G is planar and Π is hereditary and contains and excludes infinitely many planar graphs. Finally, we provide similar results for the problem variant where the deleted vertex set has to induce a connected graph.

An improved FPT algorithm for Almost Forest Deletion problem

Almost Forest Deletion problem (AFD) is a generalization of the Feedback Vertex Set problem, which decides whether there exist at most k vertices in a given graph G whose removal from G results in an l-forest, where k and l are two given non-negative integers, and an l-forest is a graph which can be transformed into a forest by deleting at most l edges. Based on the iterative compression technique, we study the iterative version of the AFD problem, called Almost Forest Deletion Disjoint Compression problem (AFDDC), which asks for a new l-forest deletion set X′ of size at most k for a given graph G that is disjoint with a given l-forest deletion set X of graph G for two given non-negative integers k and l. For the AFDDC problem, we develop some reduction rules to simplify a given instance, and give a new branching algorithm for the problem. A new branching measure is presented to evaluate the efficiency of the algorithm, which results in an algorithm of running time O⁎(4k+l). Based on the proposed algorithm for the AFDDC problem, a parameterized algorithm for the AFD problem with running time O⁎(5k4l) is presented, improving the previous result O⁎(5.0024(k+l)).

Deadlock resolution in wait-for graphs by vertex/arc deletion

A deadlock occurs in a distributed computation if a group of processes wait indefinitely for resources from each other. In this paper we study actions to be taken after deadlock detection, especially the action of searching for a small deadlock-resolution set. More precisely, given a “snapshot” graph G representing a deadlocked state of a distributed computation governed by a certain deadlock model $${\mathbb {M}}$$ , we investigate the complexity of vertex/arc deletion problems that aim at finding minimum vertex/arc subsets whose removal turns G into a deadlock-free graph (according to model $${\mathbb {M}}$$ ). Our contributions include polynomial-time algorithms and hardness proofs, for general graphs and for special graph classes. Among other results, we show that the arc deletion problem in the OR model can be solved in polynomial time, and the vertex deletion problem in the OR model remains NP-complete even for graphs with maximum degree $${\varDelta }(G) = 4$$ , but is solvable in $$O (m \sqrt{n})$$ time for graphs with $${\varDelta }(G)\le 3$$ .

Unit interval vertex deletion: Fewer vertices are relevant

The unit interval vertex deletion problem asks for a set of at most k vertices whose deletion from a graph makes it a unit interval graph. We develop an O(k4)-vertex kernel for the problem, significantly improving the O(k53)-vertex kernel of Fomin et al. (2013) [11]. We start from a constant-approximation solution and study its interaction with other vertices, which induce a unit interval graph. We partition vertices of this unit interval graph into cliques in a naive way, and pick a small number of representatives from each of them. Our constructive proof for the correctness of our algorithm, using interval models, greatly simplifies the “destructive” proofs, based on destroying forbidden structures, for similar problems in literature. Our algorithm can be implemented in O(mn+n2) time, where n and m denote respectively the numbers of vertices and edges of the input graph.

Parameterised Algorithms for Deletion to Classes of DAGs

In the Directed Feedback Vertex Set (DFVS) problem, we are given a digraph D on n vertices and a positive integer k, and the objective is to check whether there exists a set of vertices S such that F = D − S is an acyclic digraph. In a recent paper, Mnich and van Leeuwen [STACS 2016 ] studied the kernelization complexity of DFVS with an additional restriction on F—namely that F must be an out-forest, an out-tree, or a (directed) pumpkin—with an objective of shedding some light on the kernelization complexity of the DFVS problem, a well known open problem in the area. The vertex deletion problems corresponding to obtaining an out-forest, an out-tree, or a (directed) pumpkin are Out-forest/Out-tree/Pumpkin Vertex Deletion Set, respectively. They showed that Out-forest/Out-tree/Pumpkin Vertex Deletion Set admit polynomial kernels. Another open problem regarding DFVS is that, does DFVS admit an algorithm with running time $2^{\mathcal {O}(k)} n^{\mathcal {O}(1)}$ ? We complement the kernelization programme of Mnich and van Leeuwen by designing fast FPT algorithms for the above mentioned problems. In particular, we design an algorithm for Out-forest Vertex Deletion Set that runs in time $\mathcal {O}(2.732^{k} n^{\mathcal {O}(1)})$ and algorithms for Pumpkin/Out-tree Vertex Deletion Set that runs in time $\mathcal {O}(2.562^{k} n^{\mathcal {O}(1)})$ . As a corollary of our FPT algorithms and the recent result of Fomin et al. [STOC 2016] which gives a relation between FPT algorithms and exact algorithms, we get exact algorithms for Out-forest/Out-tree/Pumpkin Vertex Deletion Set that run in time $\mathcal {O}(1.633^{n} n^{\mathcal {O}(1)})$ , $\mathcal {O}(1.609^{n} n^{\mathcal {O}(1)})$ and $\mathcal {O}(1.609^{n} n^{\mathcal {O}(1)})$ , respectively.

PTAS for [formula omitted]-free node deletion problems in disk graphs

For a set H of graphs, a graph G is H-free if G does not contain any subgraph isomorphic to some graph in H. In this paper, we study the minimum H-free node deletion problem (MinHFND) and the maximum H-free node set problem (MaxHFNS), which include a lot of extensively-studied problems such as the minimum k-path vertex cover problem, the dissociation number problem, and the minimum degree bounded node deletion problem. For a large class of H, PTASs are given for MinHFND and MaxHFNS on disk graphs whose heterogeneity is bounded by a constant, where the heterogeneity of a disk graph is the ratio of the maximum radius to the minimum radius of disks.

Approximation and Kernelization for Chordal Vertex Deletion

The Chordal Vertex Deletion (ChVD) problem asks to delete a minimum number of vertices from an input graph to obtain a chordal graph. In this paper we develop a polynomial kernel for ChVD under the parameterization by the solution size. Using a new Erdös--Pósa-type packing/covering duality for holes in nearly chordal graphs, we present a polynomial-time algorithm that reduces any instance $(G,k)$ of ChVD to an equivalent instance with ${poly}(k)$ vertices. The existence of a polynomial kernel answers an open problem posed by Marx in 2006 [D. Marx, “Chordal Deletion Is Fixed-Parameter Tractable,” in Graph-Theoretic Concepts in Computer Science, Lecture Notes in Comput. Sci. 4271, Springer, 2006, pp. 37--48]. To obtain the kernelization, we develop the first ${poly}({\sc opt})$-approximation algorithm for ChVD, which is of independent interest. In polynomial time, it either decides that $G$ has no chordal deletion set of size $k$, or outputs a solution of size $\mathcal{O}(k^4\log^2k)$.

A Polynomial Kernel for Trivially Perfect Editing

We give a kernel with O(k^7) vertices for Trivially Perfect Editing, the problem of adding or removing at most k edges in order to make a given graph trivially perfect. This answers in affirmative an open question posed by Nastos and Gao (Soc Netw 35(3):439–450, 2013), and by Liu et al. (Tsinghua Sci Technol 19(4):346–357, 2014). Our general technique implies also the existence of kernels of the same size for related Trivially Perfect Completion and Trivially Perfect Deletion problems. Whereas for the former an O(k^3) kernel was given by Guo (in: ISAAC 2007, LNCS, vol 4835, Springer, pp 915–926, 2007), for the latter no polynomial kernel was known. We complement our study of Trivially Perfect Editing by proving that, contrary to Trivially Perfect Completion, it cannot be solved in time 2^{o(k)}cdot n^{O(1)} unless the exponential time hypothesis fails. In this manner we complete the picture of the parameterized and kernelization complexity of the classic edge modification problems for the class of trivially perfect graphs.

Bivariate complexity analysis of Almost Forest Deletion

In this paper we study a generalization of classic Feedback Vertex Set problem in the realm of multivariate complexity analysis. We say that a graph F is an l-forest if we can delete at most l edges from F to get a forest. That is, F is at most l edges away from being a forest. In this paper we introduce the Almost Forest Deletion problem, where given a graph G and integers k and l, the question is whether there exists a subset of at most k vertices such that its deletion leaves us an l-forest. We show that this problem admits an algorithm with running time 2O(k+l)nO(1) and a kernel of size O(kl(k+l)). We also show that the problem admits a 2O(tw)nO(1) algorithm on bounded treewidth graphs, using which we design a subexponential algorithm for the problem on planar graphs.

Block Crossings in Storyline Visualizations

Storyline visualizations help visualize encounters of the characters in a story over time. Each character is represented by an x-monotone curve that goes from left to right visualizing progression of time. A meeting is represented by having the characters that participate in the meeting run close together for some time. In order to keep the visual complexity low, rather than just minimizing pairwise crossings of curves, we propose to count block crossings, that is, pairs of intersecting bundles of lines. In a block crossing, two blocks of parallel lines intersect each other, which is less distracting than the same number of individual crossings being spread over the drawing. In this paper, we show that minimizing the number of block crossings is NP-hard, even if all meetings are of size 2. For this special case, we present a greedy heuristic, which we evaluate experimentally. We show that the general case is fixed-parameter tractable. Our main results is a constant-factor approximation algorithm for meetings of bounded size. The algorithm is based on (approximately) solving a hyperedge deletion problem on hypergraphs that may be of independent interest.

Parameterized algorithms for Edge Biclique and related problems

Maximum Edge Biclique and related problems have wide applications in management science, bioinformatics, etc. In this paper, we study parameterized algorithms for Parameterized Edge Biclique problem, Parameterized Edge Biclique Packing problem, Parameterized Biclique Edge Deletion problem, and Parameterized Bipartite Biclique Clustering problem. For the Parameterized Edge Biclique problem, the current best result is of running time O⁎(2k), and we give a parameterized algorithm of running time O(nk2.5k⌈k⌉), where k is the parameter and n is the number of vertices in the given graph. For the Parameterized Edge Biclique Packing problem, based on randomized divide-and-conquer technique, a parameterized algorithm of running time O⁎(k⌈k⌉logk4(2k−1)t)) is given, where k is the parameter and t is the number of bicliques in the solution. We study the Parameterized Biclique Edge Deletion problem on bipartite graphs and general graphs, and give parameterized algorithms of running time O⁎(2k) and O⁎(3k), respectively. For the Parameterized Bipartite Biclique Clustering problem, based on modular decomposition method, a kernel of size O(k2) and a parameterized algorithm of running time O(2.42k(n+m)) are presented, where k is the parameter, n is the number of vertices, and m is the number of edges in the given graph.