Vertex Cover Research Articles

Fixed-Parameter Algorithms for Computing Bend-Restricted RAC Drawings of Graphs

In a right-angle crossing (RAC) drawing of a graph, each edge is represented as a polyline and edge crossings must occur at an angle of exactly $90^\circ$, where the number of bends on such polylines is typically restricted in some way. While structural and topological properties of RAC drawings have been the focus of extensive research and particular attention has been paid to RAC drawings with at most $0$, $1$, $2$ and $3$ bends per edge, little was known about the boundaries of tractability for computing such drawings. In this paper, we initiate the study of bend-restricted RAC drawings from the viewpoint of parameterized complexity. In particular, we establish that computing a RAC drawing of an input graph $G$ with at most $b$ bends where each edge $e$ has a prescribed upper-bound $0\leq \beta(e) \leq 3$ on the number of bends it can support (or determining that none exists) is: fixed-parameter tractable parameterized by the feedback edge number of $G$, and fixed-parameter tractable parameterized by $b$ plus the vertex cover number of $G$.

Fixed-Ratio Approximation Algorithm for the Minimum Cost Cover of a Digraph by Bounded Number of Cycles

We consider polynomial time approximation for the minimum cost cycle cover problem of an edge-weighted digraph, where feasible covers are restricted to have at most k disjoint cycles. In the literature this problem is referred to as Min-k-SCCP. The problem is closely related to classic Traveling Salesman Problem (TSP) and Vehicle Routing Problem (VRP) and has many important applications in algorithms design and operations research. Unlike its unconstrained variant, the Min-k-SCCP is strongly NP-hard even on undirected graphs and remains intractable in very specific settings. For any metric, the problem can be approximated in polynomial time within ratio 2, while in fixed-dimensional Euclidean spaces it admits Polynomial Time Approximation Schemes (PTAS). In the same time, approximation of the more general asymmetric Min-k-SCCP still remains weakly studied. In this paper, we propose the first fixed-ratio approximation algorithm for this problem, which extends the recent breakthrough Svensson-Tarnawski-Vegh and Traub-Vygen results for the Asymmetric Traveling Salesman Problem.

Digital-analog counterdiabatic quantum optimization with trapped ions

Abstract We introduce a hardware-specific, problem-dependent digital-analog quantum algorithm of a counterdiabatic quantum dynamics tailored for optimization problems. Specifically, we focus on trapped-ion architectures, taking advantage from global Mølmer–Sørensen gates as the analog interactions complemented by digital gates, both of which are available in the state-of-the-art technologies. We show an optimal configuration of analog blocks and digital steps leading to a substantial reduction in circuit depth compared to the purely digital approach. This implies that, using the proposed encoding, we can address larger optimization problem instances, requiring more qubits, while preserving the coherence time of current devices. Furthermore, we study the minimum gate fidelity required by the analog blocks to outperform the purely digital simulation, finding that it is below the best fidelity reported in the literature. To validate the performance of the digital-analog encoding, we tackle the maximum independent set problem, showing that it requires fewer resources compared to the digital case. This hybrid co-design approach paves the way towards quantum advantage for efficient solutions of quantum optimization problems.

XNLP-Completeness for Parameterized Problems on Graphs with a Linear Structure

AbstractIn this paper, we showcase the class XNLP as a natural place for many hard problems parameterized by linear width measures. This strengthens existing W[1]-hardness proofs for these problems, since XNLP-hardness implies W[t]-hardness for all t. It also indicates, via a conjecture by Pilipczuk and Wrochna (ACM Trans Comput Theory 9:1–36, 2018), that any XP algorithm for such problems is likely to require XP space. In particular, we show XNLP-completeness for natural problems parameterized by pathwidth, linear clique-width, and linear mim-width. The problems we consider are Independent Set, Dominating Set, Odd Cycle Transversal, (q-)Coloring, Max Cut, Maximum Regular Induced Subgraph, Feedback Vertex Set, Capacitated (Red-Blue) Dominating Set, Capacitated Vertex Cover and Bipartite Bandwidth.

Analyzing Single-Valued Neutrosophic Fuzzy Graphs Through Matroid Perspectives

We hope to present this paper on the emergence of a novel category of matroids derived from single-valued neutrosophic (SVN) fuzzy-graphs. The findings of this study make a substantial contribution to both matroid theory and the field of neutrosophic systems. In this study, we aim to introduce the concept of SVN-matroids and conduct a thorough investigation into their fundamental properties. Additionally, we explore the essential requirements for establishing duality within the realm of matroids formed from neutrosophic fuzzy-networks. This investigation encompasses co-weak isomorphism, weak isomorphism, and isomorphism between two matroids. Furthermore, our research proposes the expansion of other graph products and the creation of a SVN-graph to utilize the Greedy and Kruskal algorithms for determining optimal productivity in an information flow of a single-source network within a SVN-network. The application of neutronosophic graph theory offers a paradigm for addressing optimization problems in chemical graph theory, such as identifying the maximal independent set and a minimum spanning tree. When faced with uncertainty, neutrosophic optimization provides more reliable solutions by considering the uncertainties associated with the objective function or constraints.

Efficient identification of maximum independent sets in stochastic multilayer graphs with learning automata

Investigating the maximum independent set in stochastic multilayer graphs provides critical insights into the structural and dynamical properties of complex networks. Recently, stochastic multilayer graphs effectively model the intricate interactions and interdependencies inherent in real-world systems, including social, biological, and transportation networks. The identification of a maximum independent set -comprising nodes without direct connections- offers a significant understanding of phenomena such as information diffusion, resource allocation, and epidemic spread within complex social networks. For instance, independent sets play a crucial role in identifying influencers -individuals who profoundly impact their peers, propagating information or opinions widely. In this paper, we introduce the stochastic version of the maximum independent set and propose five algorithms based on learning automata to identify maximum independent sets in the stochastic multilayer graphs. Our approach utilizes learning automata to provide a guided sampling from candidate independent sets of the stochastic multilayer graph, aiming to identify the independent set with the maximum expected value while utilizing fewer vertex samples than standard methods that do not incorporate the learning. In addition to proving several mathematical properties of the proposed approach, simulations conducted across diverse stochastic multilayer graphs demonstrate that our learning automata-based algorithms outperform traditional approaches, achieving higher convergence rates and requiring fewer samples.

On estimation of extremal entries of the principal eigenvector of a graph

Let X = ( x 1 , x 2 , … , x n ) T be the principal eigenvector corresponding to the spectral radius λ 1 of a graph G of order n. In this paper, we find some bounds on the ratio of the maximal component x max to the minimal component x min of the principal eigenvector X in terms of the graph parameters such as the independence number β , the minimum vertex cover number of the vertex α and the chromatic number χ . Also, we present some bounds on the extremal component x max of the principal eigenvector X. An upper bound of the spectral radius λ 1 of G in terms of the minimum vertex cover number α and order of the graph n is also introduced in this paper.

Approximation algorithms for minimum ply covering of points with unit squares and unit disks

Given a set P of points and a set U of geometric objects in the Euclidean plane, a minimum ply cover of P with U is a subset of U that covers P and minimizes the number of objects that share a common intersection, called the minimum ply cover number of P with U. Biedl et al. (2021) [9] showed that for both unit squares and unit disks, determining the minimum ply cover number for a set of points is NP-hard. They gave polynomial-time 2-approximation algorithms for the special case when the minimum ply cover number is constant, and asked whether there exists polynomial-time O(1)-approximation algorithms for these problems. In this paper, we settle the question posed by Biedl et al. by providing polynomial-time O(1)-approximation algorithms for the minimum ply cover problem for both unit squares and unit disks.

Some properties of precompletely and positively numbered sets

In this paper, we prove a joint generalization of Arslanov's completeness criterion and Visser's ADN theorem for precomplete numberings which, for the Gödel numbering x↦Wx, has been proved by Terwijn (2018). The question of whether this joint generalization takes place in each precomplete numbering has been raised in his joint paper with Barendregt in 2019. Then we consider the properties of completeness and precompleteness of numberings in the context of the positivity property. We show that no completion of a positive numbering is a minimal cover of that numbering, and that the Turing completeness of any set A is equivalent to the existence of a positive precomplete A-computable numbering of any infinite family with positive A-computable numbering. In addition, we prove that each Σn0-computable numbering (n⩾2) of a Σn0-computable non-principal family has a Σn0-computable minimal cover ν such that for every computable function f there exists an integer n with ν(f(n))=ν(n).

Farmland Degradation in the Czech Republic: Drivers and Barriers of Mitigation Strategies in Agricultural Soils

ABSTRACTIn line with Sustainable Development Goal (SDG) 15.3, which aims to restore degraded land and soil, and the priorities of the Common Agriculture Policy (CAP) to mitigate environmental issues caused by intensive agriculture, understanding the factors influencing farmers' adoption of soil conservation practices (SCPs) is crucial. However, there is still a limited understanding of these specific factors, particularly in the context of the Czech Republic. This study investigates the perceptions and determinants influencing the adoption of SCPs among farmers in the Czech Republic. We analyzed 358 randomly selected farm households using probit and multivariate probit models. Our results show that a significant number of farmers perceive soil degradation as an important problem, attributing it to factors such as low soil nutrient content, declining soil humus, water and wind erosion. The results of our analysis showed positive associations for perceived effectiveness and profitability, indicating that farmers tend to adopt practices such as minimum tillage, mulching, mixed cropping, cover cropping, and continuous soil cover when these methods are perceived to be effective and profitable. Conversely, negative associations are found for perceptions of soil degradation and education, suggesting potential barriers to adoption with higher levels of perceived soil degradation and education. The study also highlights the complex interplay of information sources on adoption, with both positive and negative trends. In light of these findings, we propose recommendations, including that awareness campaigns should be tailored to address perceptions of soil degradation, and that the use of peer networks and information dissemination from research institutions can bridge the gap between scientific recommendations and on‐farm practices. Policy makers and agricultural extension services should work together to develop targeted strategies that take into account regional differences in the factors influencing adoption and ultimately promote widespread adoption of SCPs.

The upper geodetic vertex covering number of a graph

A set S ⊆ V (G) is a geodetic vertex cover of G if S is both a geodetic set and a vertex cover of G. The minimum cardinality of a geodetic vertex cover of G is defined as the geodetic vertex covering number of G and is denoted by gα(G) . A geodetic vertex cover S in a connected graph G is called a minimal geodetic vertex cover of G if no proper subset of S is a geodetic vertex cover of G. The upper geodetic vertex covering number g+­­α(G) of G is the maximum cardinality of a minimal geodetic vertex cover of G. Some general properties satisfied by the upper geodetic vertex covering number of a graph are studied. The upper geodetic vertex covering number of several classes of graphs are determined. Some bounds for g+­­α(G) are obtained and the graphs attaining these bounds are characterized.

Approximation algorithm for prize-collecting vertex cover with fairness constraints

Approximation algorithm for prize-collecting vertex cover with fairness constraints

Fractional Vertex Cover Reliability of Graphs

Let G be a graph and let 0 ≤ p , q and p + q ≤ 1 . Suppose that each vertex of G gets a weight of 1 with probability p , 1 2 with probability q , and 0 with probability 1 − p − q , and vertex weight probabilities are independent. The \textit{fractional vertex cover reliability} of G , denoted by FRel ( G ; p , q ) , is the probability that the sum of weights at the end-vertices of every edge in G is at least 1 . In this article, we first provide various computational formulas for FRel ( G ; p , q ) considering general graphs, basic graphs, and graph operations. Secondly, we determine the graphs which maximize FRel ( G ; p , q ) for all values of p and q in the classes of trees, connected unicyclic and bicyclic graphs with fixed order, and determine the graphs which minimize it in the classes of trees and connected unicyclic graphs with fixed order. Our results on optimal graphs extend some known results in the literature about independent sets, and the tools we developed in this paper have the potential to solve the optimality problem in other classes of graphs as well.

Minimal cover of high-dimensional chaotic attractors by embedded recurrent patterns

We propose a general method for constructing a minimal cover of high-dimensional chaotic attractors by embedded unstable recurrent patterns. By minimal cover we mean a subset of available patterns such that the approximation of chaotic dynamics by a minimal cover with a predefined proximity threshold is as good as the approximation by the full available set. The proximity measure, based on the concept of a directed Hausdorff distance, can be chosen with considerable freedom and adapted to the properties of a given chaotic system. In the context of a spatiotemporally chaotic attractor of the Kuramoto–Sivashinsky system on a periodic domain, we demonstrate that the minimal cover can be faithfully constructed even when the proximity measure is defined within a subspace of dimension much smaller than the dimension of space containing the attractor. We discuss how the minimal cover can be used to provide a reduced description of the attractor structure and the dynamics on it.

Parameterized approximation algorithms for weighted vertex cover

A vertex cover of a graph is a set of vertices of the graph such that every edge has at least one endpoint in it. In this work, we study Weighted Vertex Cover with solution size as a parameter. Formally, in the (k,W)-Vertex Cover problem, given a graph G, an integer k, a positive rational W, and a weight function w:V(G)→Q+, the question is whether G has a vertex cover of size at most k of weight at most W, with k being the parameter. An (a,b)-bi-criteria approximation algorithm for (k,W)-Vertex Cover either produces a vertex cover S such that |S|≤ak and w(S)≤bW, or decides that there is no vertex cover of size at most k of weight at most W. We obtain the following results.•A simple (2,2)-bi-criteria approximation algorithm for (k,W)-Vertex Cover in polynomial time by modifying the standard LP-rounding algorithm.•A simple exact parameterized algorithm for (k,W)-Vertex Cover running in O⁎(1.4656k) time1.•A (1+ϵ,2)-approximation algorithm for (k,W)-Vertex Cover running in O⁎(1.4656(1−ϵ)k) time.•A (1.5,1.5)-approximation algorithm for (k,W)-Vertex Cover running in O⁎(1.414k) time.•A (2−δ,2−δ)-approximation algorithm for (k,W)-Vertex Cover running in O⁎(∑i=δk(1−2δ)1+2δδk(1−2δ)2δ(δk+iδk−2iδ1−2δ)) time for any δ<0.5. For example, for (1.75,1.75) and (1.9,1.9)-approximation algorithms, we get running times of O⁎(1.272k) and O⁎(1.151k) respectively.Our algorithms (expectedly) do not improve upon the running times of the existing algorithms for the unweighted version of Vertex Cover. When compared to algorithms for the weighted version, our algorithms are the first ones to the best of our knowledge which work with arbitrary weights, and they perform well when the solution size is much smaller than the total weight of the desired solution.

Some Algorithmic Results for Eternal Vertex Cover Problem in Graphs

The eternal vertex cover problem is a variant of the vertex cover problem. It is a two-player (attacker and defender) game in which, given a graph $G=(V,E)$, the defender needs to allocate guards at some vertices so that the allocated vertices form a vertex cover. The attacker can attack one edge at a time, and the defender needs to move the guards along the edges such that at least one guard moves through the attacked edge and the new configuration still remains a vertex cover. The attacker wins if no such move exists for the defender. The defender wins if there exists a strategy to defend the graph against infinite sequence of attacks. The minimum number of guards with which the defender can form a winning strategy is called the eternal vertex cover number of $G$, and is denoted by $evc(G)$. Given a graph $G$, the problem of finding the eternal vertex cover number is NP-hard for general graphs and remains NP-hard even for bipartite graphs. We have given a polynomial time algorithm to find the Eternal vertex cover number in chain graphs and $P_4$-sparse graphs. We have also given a linear-time algorithm to find the eternal vertex cover number for split graphs, an important subclass of chordal graphs.

Structural Parameterizations for Two Bounded Degree Problems Revisited

We revisit two well-studied problems, Bounded Degree Vertex Deletion and Defective Coloring , where the input is a graph G and a target degree Δ, and we are asked either to edit or partition the graph so that the maximum degree becomes bounded by Δ. Both problems are known to be parameterized intractable for the most well-known structural parameters, such as treewidth. We revisit the parameterization by treewidth, as well as several related parameters and present a more fine-grained picture of the complexity of both problems. In particular, we present the following: — Both problems admit straightforward DP algorithms with table sizes \((\Delta +2)^\mathrm{tw}\) and \((\chi _\mathrm{d}(\Delta +1))^{\mathrm{tw}}\) , respectively, where tw is the input graph’s treewidth and \(\chi _\mathrm{d}\) the number of available colors. We show that, under the SETH, both algorithms are essentially optimal, for any non-trivial fixed values of \(\Delta , \chi _\mathrm{d}\) , even if we replace treewidth by pathwidth. Along the way, we obtain an algorithm for Defective Coloring with complexity quasi-linear in the table size, thus settling the complexity of both problems for treewidth and pathwidth. — Given that the standard DP algorithm is optimal for treewidth and pathwidth, we then go on to consider the more restricted parameter tree-depth. Here, previously known lower bounds imply that, under the ETH, Bounded Vertex Degree Deletion and Defective Coloring cannot be solved in time \(n^{o(\sqrt [4]{\mathrm{td}})}\) and \(n^{o(\sqrt {\mathrm{td}})}\) , respectively, leaving some hope that a qualitatively faster algorithm than the one for treewidth may be possible. We close this gap by showing that neither problem can be solved in time \(n^{o(\mathrm{td})}\) , under the ETH, by employing a recursive low tree-depth construction that may be of independent interest. — Finally, we consider a structural parameter that is known to be restrictive enough to render both problems FPT: vertex cover. For both problems the best known algorithm in this setting has a super-exponential dependence of the form \(\mathrm{vc}^{\mathcal {O}(\mathrm{vc})}\) . We show that this is optimal, as an algorithm with dependence of the form \(\mathrm{vc}^{o(\mathrm{vc})}\) would violate the ETH. Our proof relies on a new application of the technique of d -detecting families introduced by Bonamy et al. [ToCT 2019]. Our results, although mostly negative in nature, paint a clear picture regarding the complexity of both problems in the landscape of parameterized complexity, since in all cases we provide essentially matching upper and lower bounds.

A self‐stabilizing distributed algorithm for the 1‐MIS problem under the distance‐3 model

SummaryFault‐tolerance and self‐organization are critical properties in modern distributed systems. Self‐stabilization is a class of fault‐tolerant distributed algorithms which has the ability to recover from any kind and any finite number of transient faults and topology changes. In this article, we propose a self‐stabilizing distributed algorithm for the 1‐MIS problem under the unfair central daemon assuming the distance‐3 model. Here, in the distance‐3 model, each process can refer to the values of local variables of processes within three hops. Intuitively speaking, the 1‐MIS problem is a variant of the maximal independent set (MIS) problem with improved local optimizations. The time complexity (convergence time) of our algorithm is steps and the space complexity is bits, where is the number of processes. Finally, we extend the notion of 1‐MIS to ‐MIS for each nonnegative integer , and compare the set sizes of ‐MIS () and the maximum independent set.

Domination and Independence in Fuzzy Semigraphs

Objectives: To identify dominating and independent sets in fuzzy semigraphs by developing suitable methods. To examine dominating and independent sets in fuzzy semigraphs for their characteristics and behaviors. The goal is to examine the use of the concepts of domination and independence in the fields of analysis of social networks, security of networks, and allocation of resources. Methods: This study introduces the concept of strong arc fuzzy semigraph and explores various parameters such as dominating and independent sets. The Minimum Adjacent Dominating Number (ad-number) \gamma_a(\operatorname{𝓖}) and Maximum Adjacent Dominating Number (ad-number) \Gamma_a(\operatorname{𝓖}) , Minimum End Node Adjacent Dominating Number (ead-number) \gamma_{ea}(\operatorname{𝓖}) and Maximum End Node Adjacent Dominating Number (ead-number) \Gamma_{ae}(\operatorname{𝓖}) as well as Minimum Consecutive Adjacent Dominating Number (cad-number) \gamma_{ca}(\operatorname{𝓖}) and Maximum Consecutive Adjacent Dominating Number (cad-number) \Gamma_{ca}(\operatorname{𝓖}) . These parameters represent the respective minimum and maximum cardinalities taken over all minimal dominating sets of nodes of \operatorname{𝓖} . Additionally, it explores the e-independence dominating number i_e and e-independence number \beta_e(\operatorname{𝓖}) , strongly independence dominating number i_s and strongly independence number \beta_s(\operatorname{𝓖}) and consecutive adjacent independence dominating number i_{ca} and consecutive adjacent independence number \beta_{ca}(\operatorname{𝓖}) . These parameters represent the respective minimum and maximum cardinalities derived from all maximal independent sets of nodes in \operatorname{𝓖} . Findings: The study develops a thorough correlation between fuzzy dominant sets and independent sets in the fuzzy semigraph \operatorname{𝓖} . The statement illustrates the way in which the introduced factors interact and impact the structure and behaviour of the fuzzy semigraph. Novelty: The introduction of strong arc fuzzy semigraphs is a novel notion in the examination of fuzzy graphs. The study involves the creation and examination of novel parameters associated with dominant and independent sets in fuzzy semigraphs. This study aims to establish novel theoretical linkages between fuzzy domination and independent sets, therefore expanding our knowledge of the structural features of fuzzy semigraphs. Keywords: Domination in fuzzy semigraphs, Independent in fuzzy semigraphs, Adjacent-domination, ea-domination, ca-domination, e-independent, s-independent, ca-independent

Analytical solution of overlying pressure for shallowly buried underwater box tunnel: A case study of box jacking in Suzhou, China

Accurately predicting the overlying pressure is crucial for determining an appropriate cover depth of underwater box tunnels to avoid the uplifting failure. Based on the project of box jacking crossing the Beijing-Hangzhou Grand Canal in Suzhou, the characteristics of overlying pressure variation during tunneling are investigated. The monitoring results reveal that the fluctuation of overlying pressure is weakened during the rapid tunneling process. A modified analytical model for vertical earth pressure is conceived, in which the active and passive limit states for multi-layered soils are both considered. The probable range of overlying pressure obtained by the proposed model is suitable to cover the actual values. The anti-floating behavior of underwater box tunnels for two different working conditions is discussed by calculating the minimum cover depth. Using the calibrated analytical models, a parametric study is conducted to explore the influence of injection pressure, hardened slurry unit weight, soil internal friction angle, soil cohesion, and tunnel geometry. It is found that the injection pressure during the construction process is crucial for determining the necessary cover depth, and the change of box tunnel height makes it easier to trigger the variation of minimum cover depth.