Deletion Problem Research Articles

Modification problems toward proper (Helly) circular-arc graphs

We present a 9k⋅nO(1)-time algorithm for the proper circular-arc vertex deletion problem, resolving an open problem of van 't Hof and Villanger [Algorithmica 2013] and Crespelle et al. [Computer Science Review 2023]. Our structural study also implies parameterized algorithms for modification problems toward proper Helly circular-arc graphs.

Solving Edges Deletion Problem of Complete Graphs

مع زيادة حجم الشبكة فان احتمالية الفشل في بعض مكوناتها (العقد او الحافات) يزداد، لذلك من الضروري ان تكون الشبكة قادرة على التواصل بين مكوناتها في حالة وجود خلل في بعض مكوناتها. ان احتمالية حدوث عطل في التوصيلات بين محطات الشبكة أكبر من احتمالية حدوث عطل في المحطات نفسها بسبب طبيعة عمل العديد من الشبكات. يمكن دراسة فعالية وموثوقية الشبكة باستخدام مجموعة من تقنيات نظرية البيان. ان التواصل في الشبكة هو ما يحدد مدى صلابتها (موثوقيتها) حيث ان فعالية الشبكة تتأثر عند حدوث تلف في أحد مكوناتها. يستخدم قطر الرسم البياني لقياس كفاءة عمل الشبكة من خلال قياس اقصى تأخير في استلام إشارة (رسالة) معينة او حدوث خلل في الاستلام. في هذا البحث ندرس مدى زيادة القطر للرسم البياني الناتج بعد حذف من الحافات من الرسم البياني . وايجاد أكبر قطر للرسم بياني الذي يحتوي من الرؤوس ونحصل عليه بعد حذف من الحافات من وتحديد بعد حساب

Edges deletion problem of hypercube graphs for some n

Numerous graph-theoretic methods can be used to investigate the efficiency and reliability of networks. The reliability of a network is determined based on its connectivity. At the very least, the system must continue to function in the event of part of its component (node or link) failures. An ideal system remains reliable even when errors occur. Diameter is a measure of network efficiency used to study the effects of link failures in networks. In this paper, we look into the eliminating edges affect to the hypercube graph’s diameter (call it by [Formula: see text]) and calculate the maximum diameter that is denoted by [Formula: see text] after deleting [Formula: see text] edges from hypercube graphs [Formula: see text] for some n.

Multi-Resolution Odd Sketch for Mining Extended Jaccard Similarity of Dynamic Streaming sets

Estimating the Jaccard similarity between two or more streaming sets is a fundamental problem with many applications, such as mining the co-occurrences of host communication behaviors in IP networks and the co-purchasing behaviors of online users in E-commence systems. For a “streaming” data set, its elements arrive sequentially at the stream processing engine, whose working memory is limited in size. To meet the resource constraint, the data set must be stored as a compressed format called a “sketch”, so that millions or even billions of such sets can be held in the memory of the engine. To strike a balance between memory cost and similarity estimation accuracy, many sketching algorithms were proposed, such as MinHash, HyperLogLog and theta sketch. As far as we know, most of previous solutions fail to handle the fully dynamic streaming sets that allow both the insertion and the deletion of elements. Although PCSA <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\pm$</tex-math></inline-formula> , and virtual odd sketch (VOS) partially solves the deletion problem, their memory efficiency and estimation accuracy can be fundamentally improved. In this paper, we propose a multi-resolution odd sketch (MROS), which allows more accurate similarity estimation with less memory consumption. Its design is to encode a streaming set into multiple layers of odd sketches with exponentially reducing sampling probabilities. No matter a set is small or large, we can pick a suitable sampling probability to accurately estimate its cardinality. Next, we present an algorithm to estimate the extended Jaccard similarity of multiple streaming sets. Their compressed MROS summaries are merged by bitwise XOR, which in turn helps estimate the cardinality of symmetric difference of the multiple sets. Then, using these estimated cardinalities as observations, we estimate the size of an arbitrary set expression, which is connected by union <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\bigcup$</tex-math></inline-formula> , intersection <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\bigcap$</tex-math></inline-formula> , relative complement <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\setminus$</tex-math></inline-formula> , and symmetric difference <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\Delta$</tex-math></inline-formula> . Our evaluation results show that its accuracy outperforms MinHash, KMV, PCSA <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\pm$</tex-math></inline-formula> and VOS, and it can support element deletion.

Solving Edges Deletion Problem of Generalized Petersen Graphs

The dependability and effectiveness of a network can be investigated using a variety of graph-theoretic techniques, and the network's connectivity determines how reliable it is. Diameter in a network is often used to measure the efficiency of a network in the event that a network experiences issues such as a decline in the communication signal or a breakdown in the communication between its components. This paper examines the increase in diameter of the generalized Petersen graph after removing a certain number of edges. Finding the exact values of f(n,t) that represent the maximum diameter of an altered generalized Petersen graph GP(n,k,t) obtained after removing edges from for k=1 and t≥2.

Smaller kernels for two vertex deletion problems

In this paper we consider two vertex deletion problems. In the Block Vertex Deletion problem, the input is a graph G and an integer k, and the goal is to decide whether there is a set of at most k vertices whose removal from G result in a block graph (a graph in which every biconnected component is a clique). In the Pathwidth One Vertex Deletion problem, the input is a graph G and an integer k, and the goal is to decide whether there is a set of at most k vertices whose removal from G result in a graph with pathwidth at most one. We give a kernel for Block Vertex Deletion with O(k3) vertices and a kernel for Pathwidth One Vertex Deletion with O(k2) vertices. Our results improve the previous O(k4)-vertex kernel for Block Vertex Deletion (Agrawal et al., 2016 [1]) and the O(k3)-vertex kernel for Pathwidth One Vertex Deletion (Cygan et al., 2012 [3]).

Complexity of the (Connected) Cluster Vertex Deletion Problem on H-free Graphs

The well-known Cluster Vertex Deletion problem (cluster-vd) asks for a given graph G and an integer k whether it is possible to delete a set S of at most k vertices of G such that the resulting graph G-S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G-S$$\\end{document} is a cluster graph (a disjoint union of cliques). We give a complete characterization of graphs H for which cluster-vd on H-free graphs is polynomially solvable and for which it is NP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{NP}$$\\end{document}-complete. Moreover, in the NP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{NP}$$\\end{document}-completeness cases, cluster-vd cannot be solved in sub-exponential time in the vertex number of the H-free input graphs unless the Exponential-Time Hypothesis fails. We also consider the connected variant of cluster-vd, the Connected Cluster Vertex Deletion problem (connected cluster-vd), in which the set S has to induce a connected subgraph of G. It turns out that connected cluster-vd admits the same complexity dichotomy for H-free graphs. Our results enlarge a list of rare dichotomy theorems for well-studied problems on H-free graphs.

Algorithms for 2-club cluster deletion problems using automated generation of branching rules

In the 2-Club Cluster Vertex Deletion (resp., 2-Club Cluster Edge Deletion) problem the input is a graph G and an integer k, and the goal is to decide whether there is a set of at most k vertices (resp., edges) whose removal from G results in a graph in which the diameter of every connected component is at most 2. In this paper we give algorithms for 2-Club Cluster Vertex Deletion and 2-Club Cluster Edge Deletion whose running times are O⁎(3.104k) and O⁎(2.562k), respectively. Our algorithms were obtained using automated generation of branching rules. Our results improve the previous O⁎(3.303k)-time algorithm for 2-Club Cluster Vertex Deletion [Liu et al., FAW-AAIM 2012] and the O⁎(2.695k)-time algorithm for 2-Club Cluster Edge Deletion [Abu-Khzam et al., TCS 2023].

A simple [formula omitted]-approximation algorithm for Split Vertex Deletion

A split graph is a graph whose vertex set can be partitioned into a clique and a stable set. Given a graph G and weight function w:V(G)→Q≥0, the Split Vertex Deletion (SVD) problem asks to find a minimum weight set of vertices X such that G−X is a split graph. It is easy to show that a graph is a split graph if and only if it does not contain a 4-cycle, 5-cycle, or a two edge matching as an induced subgraph. Therefore, SVD admits an easy 5-approximation algorithm. On the other hand, for every δ>0, SVD does not admit a (2−δ)-approximation algorithm, unless P=NP or the Unique Games Conjecture fails.For every ϵ>0, Lokshtanov, Misra, Panolan, Philip, and Saurabh (Lokshtanov et al., 2020) recently gave a randomized(2+ϵ)-approximation algorithm for SVD. In this work we give an extremely simple deterministic (2+ϵ)-approximation algorithm for SVD.

Small Vertex Cover Helps in Fixed-Parameter Tractability of Graph Deletion Problems over Data Streams

In the study of parameterized streaming complexity on graph problems, the main goal is to design streaming algorithms for parameterized problems such that O(f(k)logO(1)n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}(f(k) \\log ^{\\mathcal {O}(1)} n)$$\\end{document} space is enough, where f is an arbitrary computable function depending only on the parameter k. However, in the past few years very few positive results have been established. Most of the graph problems that do have streaming algorithms of the above nature are ones where localized checking is required, like Vertex Cover or Maximum Matching parameterized by the size k of the solution we are seeking. Chitnis et al. (SODA’16) have shown that many important parameterized problems that form the backbone of traditional parameterized complexity are known to require Ω(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega (n)$$\\end{document} bits of storage for any streaming algorithm; e.g. Feedback Vertex Set, Even Cycle Transversal, Odd Cycle Transversal, Triangle Deletion or the more general F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{F}$$\\end{document}-Subgraph Deletion when parameterized by solution size k. Our contribution lies in overcoming the obstacles to efficient parameterized streaming algorithms in graph deletion problems by utilizing the power of parameterization. We focus on the vertex cover size K as the parameter for the parameterized graph deletion problems we consider. In this work, we consider the four most well-studied streaming models: the Ea, Dea, Va (vertex arrival) and Al (adjacency list) models. Surprisingly, the consideration of vertex cover size K in the different models leads to a classification of positive and negative results for problems like F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{F}$$\\end{document}-Subgraph Deletion and F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{F}$$\\end{document}-Minor Deletion.

Streaming deletion problems parameterized by vertex cover

Streaming is a model where an input graph is provided one edge at a time, instead of being able to inspect it at will. In this work, we take a parameterized approach by assuming a vertex cover of the graph is given, building on work of Bishnu et al. [COCOON 2020]. We show the further potency of combining this parameter with the Adjacency List streaming model to obtain results for vertex deletion problems. This includes kernels, parameterized algorithms, and lower bounds for the problems of Π-free Deletion, H-free Deletion, and the more specific forms of Cluster Vertex Deletion and Odd Cycle Transversal. We focus on the complexity in terms of the number of passes over the input stream, and the memory used. This leads to a pass/memory trade-off, where a different algorithm might be favourable depending on the context and instance. We also discuss implications for parameterized complexity in the non-streaming setting.

On the d-Claw Vertex Deletion Problem

On the d-Claw Vertex Deletion Problem

On the parameterized complexity of s-club cluster deletion problems

We study the parameterized complexity of the s-Club Cluster Edge Deletion (s-Club Cluster Vertex Deletion) problem: Given a graph G and two integers s≥2 and k≥1, is it possible to remove at most k edges (vertices) from G such that each connected component of the resulting graph has diameter at most s? Both s-Club Cluster Edge Deletion and s-Club Cluster Vertex Deletion problems are known to be NP-hard already when s=2. We prove that they admit a fixed-parameter tractable algorithm when parameterized by s and the treewidth of the input graph. The proof is based on a unified algorithm that solves the more general problem in which both edges and vertices can be removed from the input graph to obtain a set of disjoint components with bounded diameter. Our approach can also be exploited to solve a related problem, namely s-Club Cover, which asks whether it is possible to cover the vertices of a graph with at most d different s-clubs, for some fixed d≥1 and s≥2.

Deletion to scattered graph classes I - Case of finite number of graph classes

Graph-deletion problems involve deleting a small number of vertices so that the resulting graph belong to a given hereditary graph class. We initiate a study of a natural variation of the problem of deletion to scattered graph classes. We want to delete at most k vertices so that each connected component of the resulting graph belongs to one of the constant number of graph classes. As our main result, we show that this problem is non-uniformly fixed-parameter tractable (FPT) when the deletion problem corresponding to each of the constant number of graph classes is known to be FPT and the properties that a graph belongs to these classes are expressible in Counting Monodic Second Order (CMSO) logic. While this is shown using some black box theorems in parameterized complexity, we give a faster FPT algorithm when each of the graph classes has a finite forbidden set.

A Review of Assured Data Deletion Security Techniques in Cloud Storage

Cloud computing is an interesting technology that allows customers to have convenient, on-demand network connectivity based on their needs with minimal maintenance and contact between cloud providers. The issue of security has arisen as a serious concern, particularly in the case of cloud computing, where data is stored and accessible via the Internet from a third-party storage system. It is critical to ensure that data is only accessible to the appropriate individuals and that it is not stored in third-party locations. Because third-party services frequently make backup copies of uploaded data for security reasons, removing the data the owner submits does not guarantee the removal of the data from the cloud. Cloud data storage has grown in popularity, and the problem of ensured deletion has been solved. Several schemes to overcome the assured deletion problem have been proposed over the last few years. The proposed solutions have addressed the scaling overhead, trusted third parties, delays, single points of failure, and other inefficiencies. Customers had the option of receiving verifiable proof of deletion from cloud service providers. This article focuses on the issue of how cloud data storage clients may be confident that the deleted data from the cloud cannot be recovered. Furthermore, it discusses the practice of secure deletion. Moreover, the paper explores currently used methods to achieve the security of assuring the deletion of data faced by cloud entities such as cloud service providers, data owners, and cloud users. After that, the paper analyzes techniques to find the pros and cons of assured deletion and the problems that were solved by these techniques. Finally, the paper identifies some future directions for the development of assured deletion of cloud storage.

Deletion to scattered graph classes II - improved FPT algorithms for deletion to pairs of graph classes

The problem of deletion of vertices to a hereditary graph class is a well-studied problem in parameterized complexity. Recently, a natural extension of the problem was initiated where we are given a finite set of hereditary graph classes and we determine whether k vertices can be deleted from a given graph so that the connected components of the resulting graph belong to one of the given hereditary graph classes. The problem is shown to be fixed parameter tractable (FPT) when the deletion problem to each of the given hereditary graph classes is fixed-parameter tractable, and the property of being in any of the graph classes is expressible in the counting monodic second order (CMSO) logic. This paper focuses on pairs of specific graph classes (Π1,Π2) in which we would like the connected components of the resulting graph to belong to, and design simpler and more efficient FPT algorithms.

Polynomial Kernel for Interval Vertex Deletion

Given a graph G and an integer k , the Interval Vertex Deletion (IVD) problem asks whether there exists a subset S ⊆ V ( G ) of size at most k such that G-S is an interval graph. This problem is known to be NP -complete (according to Yannakakis at STOC 1978). Originally in 2012, Cao and Marx showed that IVD is fixed parameter tractable: they exhibited an algorithm with running time 10 k n O (1). The existence of a polynomial kernel for IVD remained a well-known open problem in parameterized complexity. In this article, we settle this problem in the affirmative.

An improved fixed-parameter algorithm for 2-Club Cluster Edge Deletion

A 2-club is a graph of diameter at most two. In the decision version of the 2-Club Cluster Edge Deletion problem, an undirected graph G is given along with an integer k≥0 as parameter, and the question is whether G can be transformed into a disjoint union of 2-clubs by deleting at most k edges. A simple fixed-parameter algorithm solves the problem in O⁎(3k), and a decade-old algorithm improved this running to O⁎(2.74k) time via a more sophisticated case analysis. Unfortunately, this latter algorithm is shown to have a flawed branching scenario. A fixed-parameter algorithm that breaks the 3k barrier is presented in this paper, with a running time in O⁎(2.692k). This also improves the previously claimed running time.

Fixed parameterized algorithms for generalized feedback vertex set problems

A graph is an r-pseudoforest if every connected component of it has a feedback edge set of size at most r. A graph is a d-quasi-forest if every connected component of it has a feedback vertex set of size at most d. The r-Pseudoforest Deletion problem (d-Quasi-Forest Deletion problem) asks to delete a minimum number of vertices to get an r-pseudoforest (a d-quasi-forest, respectively). The well-studied feedback vertex set problem is the special case of r-Pseudoforest Deletion (d-Quasi-Forest Deletion, respectively) in which r=0 (d=0, respectively).We provide an improved FPT algorithm and a smaller kernel for r-Pseudoforest Deletion when parameterized by the solution size (when r is fixed). For d-Quasi-Forest Deletion, we show that it is FPT as well when parameterized by d and the solution size.

A Cubic Vertex-Kernel for Trivially Perfect Editing

We consider the Trivially Perfect Editing problem, where one is given an undirected graph \(G = (V,E)\) and a parameter \(k \in {\mathbb {N}}\) and seeks to edit (add or delete) at most k edges from G to obtain a trivially perfect graph. The related Trivially Perfect Completion and Trivially Perfect Deletion problems are obtained by only allowing edge additions or edge deletions, respectively. Trivially perfect graphs are both chordal and cographs, and have applications related to the tree-depth width parameter and to social network analysis. All variants of the problem are known to be NP-complete (Burzyn et al., in Discret Appl Math 154(13):1824–1844, 2006; Nastos and Gao, in Soc Netw 35(3):439–450, 2013) and to admit so-called polynomial kernels (Drange and Pilipczuk, in Algorithmica 80(12):3481–3524, 2018; Guo, in: Tokuyama, (ed) Algorithms and Computation, 18th International Symposium, ISAAC. Lecture Notes in Computer Science, Springer, Sendai, 2007. https://doi.org/10.1007/978-3-540-77120-3_79; Bathie et al., in Algorithmica 1–27, 2022). More precisely, Drange and Pilipczuk (Algorithmica 80(12):3481–3524, 2018) provided \(O(k^7)\) vertex-kernels for these problems and left open the existence of cubic vertex-kernels. In this work, we answer positively to this question for all three variants of the problem. Notice that a quadratic vertex-kernel was recently obtained for Trivially Perfect Completion by Bathie et al. (Algorithmica 1–27, 2022).