Vertex Cover Research Articles

Resource efficient stabilization for local tasks despite unknown capacity links

Self-stabilizing protocols enable distributed systems to recover correct behavior starting from any arbitrary configuration. In particular, when processors communicate by message passing and the communication links are unbounded, fake messages may be placed in communication links by an adversary. When the number of such fake messages is unknown, self-stabilization may require huge resources:•generic solutions (a.k.a. data link protocols) require unbounded resources, which makes them unrealistic to deploy,•specific solutions (e.g., census or tree construction) require O(nlog⁡n) or O(Δlog⁡n) bits of memory per node, where n denotes the network size and Δ its maximum degree, which may prevent scalability.We investigate the possibility of resource-efficient self-stabilizing protocols in this context.Specifically, we present self-stabilizing protocols for (Δ+1)-coloring and maximal independent set construction in any n-node graph, under the asynchronous message-passing model. The problems of (Δ+1)-coloring and maximal independent set construction are widely regarded as benchmark problems for evaluating local algorithms. Our protocols offer many desirable features. They are deterministic, converge in O(kΔn2log⁡n) message exchanges, where k is the (unknown) initial number of (possibly corrupted) messages in a communication link. They use messages of O(log⁡log⁡n+log⁡Δ) bits with a memory of O(Δlog⁡Δ+log⁡log⁡n) bits at each node. The resource consumption of our protocols is thus almost oblivious to the number of nodes, enabling scalability. Moreover, a striking property of our protocols is that the nodes do not need to know the number, or any bound on the number of messages initially present in each communication link of the initial (potentially corrupted) network configuration. This permits our protocols to handle any future network with unknown message capacity communication links.A key building block of our coloring and maximal independent set schemes is an algorithm to obtain an acyclic orientation of graph edges, that is of independent interest, and can serve as a useful tool for solving other tasks in this challenging setting.

An Effective Biochar Application for Reducing Nitrogen Emissions from Buffalo Digestate Storage Tank

Open manure storage contributes to the release of ammonia (NH3) into the atmosphere. Tank floating covers represent an effective technique to reduce NH3 emissions and biochar has been gain attention as a floating cover and as manure additive. Nevertheless, the mechanisms involved in the process still need to be elucidated since they are influenced by the biochar specific properties, application methods and dose. This work aims to study: (i) the biochar adsorption performances in an NH3 aqueous solution under conditions relevant to manure storage and (ii) the effect of different biochar application methods and dosage on NH3 emissions from buffalo digestate storage. The results show that a 43% reduction in NH3 emissions can be achieved by using biochar as a floating cover of 2 cm rather than as an additive. Moreover, the results show that the biochar produced at 550 °C acts as an adsorbent material for both NH4+ and NH3, by being adsorbed on the biochar surface in the form of NH4+ after H+ abstraction from the acid groups. A minimum cover height of 2 cm is required to give compactness and provide an additional resistance to the gas transfer, which is even more relevant than the adsorption in reducing NH3 emissions.

Minimal cover groups

Let F be a set of finite groups. A finite group G is called an F-cover if every group in F is isomorphic to a subgroup of G. An F-cover is called minimal if no proper subgroup of G is an F-cover, and minimum if its order is smallest among all F-covers. We prove several results about minimal and minimum F-covers: for example, every minimal cover of a set of p-groups (for p prime) is a p-group (and there may be finitely or infinitely many, for a given set); every minimal cover of a set of perfect groups is perfect; and a minimum cover of a set of two nonabelian simple groups is either their direct product or simple. Our major theorem determines whether {Zq,Zr} has finitely many minimal covers, where q and r are distinct primes. Motivated by this, we say that n is a Cauchy number if there are only finitely many groups which are minimal (under inclusion) with respect to having order divisible by n, and we determine all such numbers. This extends Cauchy's theorem. We also define a dual concept where subgroups are replaced by quotients, and we pose a number of problems.

Analysis of service life and reliability of C50CASC structures in the splash zone of the South China Sea

This paper presents the results of a reliability analysis of the service life of hydraulic structures made of C50 coral aggregate seawater concrete (CASC) in the splash zone of the South China Sea. The analysis was conducted using the ChaDuraLife V1.0 software, which is based on reliability theory and modified chloride diffusion theory. Several parameters such as the surface free chloride concentration (Cs), chloride binding capacity (R), free chloride diffusion coefficient (Df), apparent chloride diffusion coefficient (Da), and time-dependent index (m) were considered in this analysis. The (1-m) boundary condition is adopted in this study. The effect of concrete cover thickness on the service life of CASC structures was discussed. The influence of rebar types and external protective measures on the service life was also investigated. Durability recommendations for C50CASC hydraulic structures in the splash zone were proposed. The minimum concrete cover thickness needed to achieve the service life of 30a to 100a was determined to provide theoretical support for the application of CASC in island and reef engineering. The results found that the best technical measures for extending the service life were the application of 2205 duplex stainless steel bars and silane protective coating on the surface of concrete structures. A concrete cover thickness of 6–7 cm could achieve the service life of 50a, while a thickness of 8.5–10 cm, could achieve the service life of 100a.

Approximating Maximum Independent Set for Rectangles in the Plane

Approximating Maximum Independent Set for Rectangles in the Plane

Optimal strategies in fractional games: vertex cover and domination

Optimal strategies in fractional games: vertex cover and domination

Approximating power node-deletion problems

In the Power Vertex Cover(PVC) problem introduced in [1] as a generalization of the well-known Vertex Cover, we are allowed to specify costs for covering edges in a graph individually. Namely, two (nonnegative) weights, w(u,v) and w(v,u), are associated with each edge {u,v}∈E of an input graph G=(V,E), and to cover an edge {u,v}, it is required to assign “power” p∈R+V on vertices of G such that either p(u)≥w(u,v) or p(v)≥w(v,u). The objective is to minimize the total power assigned on V, ∑v∈Vp(v), while covering all the edges of G by p.The node-deletion problem for a graph property π is the problem of computing a minimum vertex subset C⊆V in a given graph G=(V,E), such that the graph satisfies π when all the vertices in C are removed from G. In this paper we consider node-deletion problems extended with the “covering-by-power” condition as in PVC, and present a unified approach for effectively approximating them. The node-deletion problems considered are Partial Vertex Cover (PartVC), Bounded Degree Deletion (BDD), and Feedback Vertex Set (FVS), each corresponding to graph properties π= “the graph has at most |E|−k edges”, π= “vertex degree of v is no larger than b(v)”, and π= “the graph is acyclic”, respectively. After reducing these problems to the Submodular Set Cover (SSC) problem, we conduct an extended analysis of the approximability of these problems in the new setting of power covering by applying some of the existing techniques for approximating SSC. It will be shown that 1) PPartVC can be approximated within a factor of 2, 2) PBDD for b∈Z+V within max⁡{2,1+bmax}, where bmax=maxv∈V⁡b(v), or within 2+log⁡bmax (for bmax≥1) by a combination of the greedy SSC algorithm and the local ratio method extended for power node-deletion problems, and 3) PFVS within 2, resulting in each of these bounds matching the best one known for the corresponding original problem.

The effect of classical optimizers and Ansatz depth on QAOA performance in noisy devices

The Quantum Approximate Optimization Algorithm (QAOA) is a variational quantum algorithm for Near-term Intermediate-Scale Quantum computers (NISQ) providing approximate solutions for combinatorial optimization problems. The QAOA utilizes a quantum-classical loop, consisting of a quantum ansatz and a classical optimizer, to minimize some cost function, computed on the quantum device. This paper presents an investigation into the impact of realistic noise on the classical optimizer and the determination of optimal circuit depth for the Quantum Approximate Optimization Algorithm (QAOA) in the presence of noise. We find that, while there is no significant difference in the performance of classical optimizers in a state vector simulation, the Adam and AMSGrad optimizers perform best in the presence of shot noise. Under the conditions of real noise, the SPSA optimizer, along with ADAM and AMSGrad, emerge as the top performers. The study also reveals that the quality of solutions to some 5 qubit minimum vertex cover problems increases for up to around six layers in the QAOA circuit, after which it begins to decline. This analysis shows that increasing the number of layers in the QAOA in an attempt to increase accuracy may not work well in a noisy device.

Distributed Independent Sets in Interval and Segment Intersection Graphs

The Maximum Independent Set problem is well-studied in graph theory and related areas. An independent set of a graph is a subset of non-adjacent vertices of the graph. A maximum independent set is an independent set of maximum size. This paper studies the Maximum Independent Set problem in some classes of geometric intersection graphs in a distributed setting. More precisely, we study the Maximum Independent Set problem on two geometric intersection graphs, interval and axis-parallel segment intersection graphs, and present deterministic distributed algorithms in a model that is similar but a little weaker than the local communication model. We compute the maximum independent set on interval graphs in [Formula: see text] rounds and [Formula: see text] messages, where [Formula: see text] is the size of the maximum independent set and [Formula: see text] is the number of nodes in the graph. We provide a matching lower bound of [Formula: see text] on the number of rounds, whereas [Formula: see text] is a trivial lower bound on message complexity. Thus, our algorithm is both time and message-optimal. We also study the Maximum Independent Set problem in interval count [Formula: see text] graphs, a special case of the interval graphs where the intervals have exactly [Formula: see text] different lengths. We propose an [Formula: see text]-approximation algorithm that runs in [Formula: see text] round. For axis-parallel segment intersection graphs, we design an [Formula: see text]-approximation algorithm that obtains a solution in [Formula: see text] rounds. The results in this paper extend the results of Molla et al. [J. Parallel Distrib. Comput. 2019].

Knowledge of macrophyte requirements and tolerances holds the key to successful shallow lake restoration – a New Zealand perspective

Many shallow lakes are degraded to the point where they are permanently turbid and macrophytes no longer grow. Without the macrophytes, wave action resuspends lakebed sediments, and a feedback loop is set up, trapping the lake in a degraded state. Multiple restoration actions are required to reverse this process, including catchment management and in-lake actions to remove barriers to establishing native vegetation. In particular, when macrophytes have been lost for a long time, the native seedbank may be depauperate and limit re-establishment opportunities. Re-establishment of submerged vegetation is critical to the restoration of lakes, but to date there have been few efforts to actively do this. There are significant barriers preventing this goal, relating to poor water quality, low-density sediments and the browsing impacts of fish and waterfowl, that will be further exacerbated by the warmer temperatures and more extreme weather events forecast under climate change. A combination of experimental and field studies determining optimal environmental ranges of different macrophyte species, methods to culture en masse and techniques to promote sufficient water clarity and sediment density to sustain macrophytes are necessary. Once minimum vegetation cover thresholds are exceeded, sustainable restoration to a macrophyte-dominated clear-water state should eventuate.

Effect of constraint relaxation on dynamic critical phenomena in minimum vertex cover problem

The effects of constraint relaxation on dynamic critical phenomena in the Minimum Vertex Cover (MVC) problem on Erdős-Rényi random graphs are investigated using Markov chain Monte Carlo simulations. Following our previous work that revealed the reduction of the critical temperature by constraint relaxation based on the penalty function method, this study focuses on investigating the critical properties of the relaxation time along its phase boundary. It is found that the dynamical correlation function of MVC with respect to the problem size and the constraint strength follows a universal scaling function. The analysis shows that the relaxation time decreases as the constraints are relaxed. This decrease is more pronounced for the critical amplitude than for the critical exponent, and this result is interpreted in terms of the system's microscopic energy barriers due to the constraint relaxation.

A succinct and approximate greedy algorithm for the Minimum Set Cover Problem

The Minimum Set Cover Problem (MSCP) is a combinatorial optimization problem belonging to the NP-Hard class in computer science. For this reason, there is no algorithm that in the worst case ensures finding an optimal solution in polynomial-time. For a given universe X, the popular greedy heuristic, called Greedy-SetCover, is the main theoretical contribution to obtain an approximate solution for the MSCP in polynomial-time, offering an optimal approximate ratio of (ln|X|+1). In this article, we propose an approximate algorithm for MSCP within a succinct representation of the input dataset, whose empirical performance improves Greedy-SetCover both in quality and execution time, while offering the same optimal approximation ratio for the problem. Our experiments show that the proposed algorithm is magnitudes of times faster than the aforementioned greedy one, obtaining on average a cardinality much closer to the optimal solution. Furthermore, because we work on a succinct representation that allows us to compute operations between sets using bitwise operators, we can process much larger datasets than state-of-the-art solutions. As a result, our proposal is also a suitable alternative for processing large datasets as required by the current Big Data era.

An efficient approach for discovering Graph Entity Dependencies (GEDs)

Graph entity dependencies (GEDs) are novel graph constraints, unifying keys and functional dependencies, for property graphs. They have been found useful in many real-world data quality and data management tasks, including fact checking on social media networks and entity resolution. In this paper, we study the discovery problem of GEDs—finding a minimal cover of valid GEDs in a given graph data. We formalise the problem, and propose an effective and efficient approach to overcome major bottlenecks in GED discovery. In particular, we leverage existing graph partitioning algorithms to enable fast GED-scope discovery, and employ effective pruning strategies over the prohibitively large space of candidate dependencies. Furthermore, we define an interestingness measure for GEDs based on the minimum description length principle, to score and rank the mined cover set of GEDs. Finally, we demonstrate the scalability and effectiveness of our GED discovery approach through extensive experiments on real-world benchmark graph data sets; and present the usefulness of the discovered rules in different downstream data quality management applications.

Resilient leaderless and leader–follower consensus over random networks through [formula omitted]-hop communication

This paper studies the resilient consensus problems under ℓ-hop communication over directed random networks. We develop novel multi-hop protocols to tackle both leaderless and leader–follower consensus with a single leader when the multiagent system is under both attack from Byzantine agents and failure of communication edges. Our unified framework features a multiple-input multiple-output system design, and generalizes the weighted mean subsequence reduced algorithms, in which extremal values in the ℓ-hop neighborhood of a cooperative agent are censored based on the notion of minimum message cover. We establish conditions on dynamical network structures, under which secure consensus with vector states can be achieved in the sense of almost sure convergence.

On blockers and transversals of maximum independent sets in co-comparability graphs

In this paper, we consider the following two problems: (i) Deletion Blocker (α) where we are given an undirected graph G=(V,E) and two integers k,d≥1 and ask whether there exists a subset of vertices S⊆V with |S|≤k such that α(G−S)≤α(G)−d, that is the independence number of G decreases by at least d after having removed the vertices from S; (ii) Transversal (α) where we are given an undirected graph G=(V,E) and two integers k,d≥1 and ask whether there exists a subset of vertices S⊆V with |S|≤k such that for every maximum independent set I we have |I∩S|≥d. We show that both problems are polynomial-time solvable in the class of co-comparability graphs by reducing them to the well-known Vertex Cut problem. Our results generalise a result of Chang et al. (2001) and a recent result of Hoang et al. (2023).

Composing dynamic programming tree-decomposition-based algorithms

Given two integers $\ell$ and $p$ as well as $\ell$ graph classes $\mathcal{H}_1,\ldots,\mathcal{H}_\ell$, the problems $\mathsf{GraphPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell,p)$, \break $\mathsf{VertPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell)$, and $\mathsf{EdgePart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell)$ ask, given graph $G$ as input, whether $V(G)$, $V(G)$, $E(G)$ respectively can be partitioned into $\ell$ sets $S_1, \ldots, S_\ell$ such that, for each $i$ between $1$ and $\ell$, $G[S_i] \in \mathcal{H}_i$, $G[S_i] \in \mathcal{H}_i$, $(V(G),S_i) \in \mathcal{H}_i$ respectively. Moreover in $\mathsf{GraphPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell,p)$, we request that the number of edges with endpoints in different sets of the partition is bounded by $p$. We show that if there exist dynamic programming tree-decomposition-based algorithms for recognizing the graph classes $\mathcal{H}_i$, for each $i$, then we can constructively create a dynamic programming tree-decomposition-based algorithms for $\mathsf{GraphPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell,p)$, $\mathsf{VertPart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell)$, and $\mathsf{EdgePart}(\mathcal{H}_1, \ldots, \mathcal{H}_\ell)$. We apply this approach to known problems. For well-studied problems, like VERTEX COVER and GRAPH $q$-COLORING, we obtain running times that are comparable to those of the best known problem-specific algorithms. For an exotic problem from bioinformatics, called DISPLAYGRAPH, this approach improves the known algorithm parameterized by treewidth.

Rydberg‐Atom Graphs for Quadratic Unconstrained Binary Optimization Problems

AbstractThere is a growing interest in harnessing the potential of the Rydberg‐atom system to address complex combinatorial optimization challenges. Here an experimental demonstration of how the quadratic unconstrained binary optimization (QUBO) problem can be effectively addressed using Rydberg‐atom graphs is presented. The Rydberg‐atom graphs are configurations of neutral atoms organized into mathematical graphs, facilitated by programmable optical tweezers, and designed to exhibit many‐body ground states that correspond to the maximum independent set (MIS) of their respective graphs. Four elementary Rydberg‐atom subgraph components are developed, not only to eliminate the need of local control but also to be robust against interatomic distance errors, while serving as the building blocks sufficient for formulating generic QUBO graphs. To validate the feasibility of the approach, a series of Rydberg‐atom experiments selected to demonstrate proof‐of‐concept operations of these building blocks are conducted. These experiments illustrate how these components can be used to programmatically encode the QUBO problems to Rydberg‐atom graphs and, by measuring their many‐body ground states, how their QUBO solutions are determined subsequently.

Parameterized Complexity of Streaming Diameter and Connectivity Problems

We initiate the investigation of the parameterized complexity of Diameter and Connectivity in the streaming paradigm. On the positive end, we show that knowing a vertex cover of size k allows for algorithms in the Adjacency List (AL) streaming model whose number of passes is constant and memory is O(logn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}(\\log n)$$\\end{document} for any fixed k. Underlying these algorithms is a method to execute a breadth-first search in O(k)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}(k)$$\\end{document} passes and O(klogn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}(k \\log n)$$\\end{document} bits of memory. On the negative end, we show that many other parameters lead to lower bounds in the AL model, where Ω(n/p)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega (n/p)$$\\end{document} bits of memory is needed for any p-pass algorithm even for constant parameter values. In particular, this holds for graphs with a known modulator (deletion set) of constant size to a graph that has no induced subgraph isomorphic to a fixed graph H, for most H. For some cases, we can also show one-pass, Ω(nlogn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega (n \\log n)$$\\end{document} bits of memory lower bounds. We also prove a much stronger Ω(n2/p)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega (n^2/p)$$\\end{document} lower bound for Diameter on bipartite graphs. Finally, using the insights we developed into streaming parameterized graph exploration algorithms, we show a new streaming kernelization algorithm for computing a vertex cover of size k. This yields a kernel of 2k vertices (with O(k2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}(k^2)$$\\end{document} edges) produced as a stream in poly(k)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {poly}(k)$$\\end{document} passes and only O(klogn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}(k \\log n)$$\\end{document} bits of memory.

Reevaluating the Rythu Bandhu Scheme: Toward Sustainable and Inclusive Agriculture in Telangana: A Review

The Rythu Bandhu Scheme (RBS), initiated by the Government of Telangana, India, on February 25, 2018, provides financial assistance to farmers across all categories. Its primary objective is to prevent farmers from succumbing to debt and to improve their financial stability. By enabling them to procure essential inputs like fertilizers, seeds, machinery, and labour, the scheme aims to enhance effective agricultural practices. However, it falls short in addressing critical environmental and social challenges in agriculture, including soil degradation, water pollution, climate change and food security. This paper proposes a comprehensive set of recommendations to enhance the Rythu Bandhu Scheme (RBS), ensuring its long-term sustainability and inclusivity for Telangana’s agricultural sector. The following strategies are crucial for achieving these goals: To enhance soil health and reduce erosion, encourage practices like minimum tillage, crop rotation and cover cropping. Livestock rearing provides supplementary income, valuable manure and draught power, diversifying farmers’ livelihoods. Address environmental and health concerns related to chemical fertilizers by recommending organic or bio-fertilizers. Utilize crop residues as mulch, fodder, or biofuel instead of burning them to minimize air pollution and greenhouse gas emissions. Establish vegetation strips along water bodies to safeguard water quality, preserve biodiversity, and prevent soil erosion. Financial incentives for sustainable practices ensure farmers’ profitability and food security. Implement district-level advisory bodies to educate farmers about sustainable methods. These recommendations align with the sustainable development goals (SDGs) agenda, benefiting both farmers and the environment.

Getting linear time in graphs of bounded neighborhood diversity

AbstractParameterized complexity, introduced to efficiently solve NP‐hard problems for small values of a fixed parameter, has been recently used as a tool to speed up algorithms for tractable problems. Following this line of research, we design algorithms parameterized by neighborhood diversity () for several graph theoretic problems in : Maximum ‐Matching, Triangle Counting and Listing, Girth, Global Minimum Vertex Cut, and Perfect Graphs Recognition. Such problems are known to admit algorithms parameterized by modular‐width () and consequently—as is a special case of —by . However, the proposed novel algorithms allow for improving the computational complexity from time —where and denote, respectively, the number of vertices and edges in the input graph—to time which is only additive in the size of the input. Then we consider some classical NP‐hard problems (Maximum independent set, Maximum clique, and Minimum dominating set) and show that for several classes of hereditary graphs, they admit linear time algorithms for sufficiently small—nonnecessarily constant—values of the neighborhood diversity parameter.