Vertex Cover Research Articles

Evolutionary Algorithm on General Cover with Theoretically Guaranteed Approximation Ratio

Theoretical studies on evolutionary algorithms have developed vigorously in recent years. Many such algorithms have theoretical guarantees in both running time and approximation ratio. Some approximation mechanism seems to be inherently embedded in many evolutionary algorithms. In this paper, we identify such a relation by proposing a unified analysis framework for a global simple multiobjective evolutionary algorithm (GSEMO) and apply it on a minimum weight general cover problem, which is general enough to subsume many important problems including the minimum submodular cover problem in which the submodular function is real-valued, and the minimum connected dominating set problem for which the potential function is nonsubmodular. We show that GSEMO yields theoretically guaranteed approximation ratios matching those achievable by a greedy algorithm in expected polynomial time when the potential function g is polynomial in the input size and the minimum gap between different g-values is a constant. History: Accepted by Erwin Pesch, Area Editor for Heuristic Search & Approximation Algorithms. Funding: This work was supported by National Natural Science Foundation of China [11771013, U20A2068]; Zhejiang Provincial Natural Science Foundation of China [LD19A010001].

A Dynamically Programmable Quantum Photonic Microprocessor for Graph Computation

AbstractQuantum computing has grown extensively, especially in system design and development, and the current research focus has gradually evolved from validating quantum advantage to practical applications. In particular, nondeterministic‐polynomial‐time (NP)‐complete problems are central in numerous important application areas. Still, in practice, it is difficult to solved efficiently with conventional computers, limited by the exponential jump in hardness. Here, a quantum photonic microprocessor based on Gaussian boson sampling (GBS) that offers dynamic programmability to solve various graph‐related NP‐complete problems is demonstrated. The system with optical, electrical, and thermal packaging implements a GBS with 16 modes of single‐mode squeezed vacuum states, a universal programmable 16‐mode interferometer, and a single photon readout on all outputs with high accuracy, generality, and controllability. The developed system is applied to demonstrate applications in solving NP‐complete problems, manifesting the ability of photonic quantum computing to realize practical applications for conventionally intractable computations. The GBS‐based quantum photonic microprocessor is applied to solve task assignment, Boolean satisfiability, graph clique, max cut, and vertex cover. These demonstrations suggest an excellent benchmarking platform, paving the way toward large‐scale combinatorial optimization.

A polytime preprocess algorithm for the maximum independent set problem

A polytime preprocess algorithm for the maximum independent set problem

Computing well-covered vector spaces of graphs using modular decomposition

A graph is well-covered if all its maximal independent sets have the same cardinality. This concept was introduced by Plummer in 1970 and naturally generalizes to the weighted case. Given a graph G, a real-valued vertex weight function w is said to be a well-covered weighting of G if all its maximal independent sets are of the same weight with respect to w. The set of all well-covered weightings of a graph G forms a vector space over the field of real numbers, called the well-covered vector space of G. Since the problem of recognizing well-covered graphs is co\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{co}$$\\end{document}-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, the problem of computing the well-covered vector space of a given graph is co\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf{co}$$\\end{document}-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}-hard. Levit and Tankus showed in 2015 that the problem admits a polynomial-time algorithm in the class of claw-free graphs. In this paper, we give two general reductions for the problem, one based on anti-neighborhoods and one based on modular decomposition, combined with Gaussian elimination. Building on these results, we develop a polynomial-time algorithm for computing the well-covered vector space of a given fork-free graph, generalizing the result of Levit and Tankus. Our approach implies a polynomial-time recognition algorithm for the class of well-covered fork-free graphs and also generalizes some known results on cographs.

Eventually lattice-linear algorithms

Lattice-linear systems allow nodes to execute asynchronously. We introduce eventually lattice-linear algorithms, where lattices are induced only among the states in a subset of the state space. The algorithm guarantees that the system transitions to a state in one of the lattices. Then, the algorithm behaves lattice linearly while traversing to an optimal state through that lattice.We present a lattice-linear self-stabilizing algorithm for service demand based minimal dominating set (SDMDS) problem. Using this as an example, we elaborate the working of, and define, eventually lattice-linear algorithms. Then, we present eventually lattice-linear self-stabilizing algorithms for minimal vertex cover (MVC), maximal independent set (MIS), graph colouring (GC) and 2-dominating set problems (2DS).Algorithms for SDMDS, MVC and MIS converge in 1 round plus n moves (within 2n moves), GC in n+4m moves, and 2DS in 1 round plus 2n moves (within 3n moves). These results are an improvement over the existing literature. We also present experimental results to show performance gain demonstrating the benefit of lattice-linearity.

Advice complexity of adaptive priority algorithms

The priority model was introduced to capture “greedy-like” algorithms. Motivated by the success of advice complexity in the area of online algorithms, the fixed priority model was extended to include advice, and a reduction-based framework was developed for proving lower bounds on the amount of advice required to achieve certain approximation ratios in this rather powerful model. To capture most of the algorithms that are considered greedy-like, the even stronger model of adaptive priority algorithms is needed. We extend the adaptive priority model to include advice. We modify the reduction-based framework from the fixed priority case to work with the more powerful adaptive priority algorithms, simplifying the proof of correctness and strengthening all previous lower bounds by a factor of two in the process.As evidence that adding advice to adaptive priority algorithms extends both adaptive priority algorithms and online algorithms with advice, we present a purely combinatorial adaptive priority algorithm with advice for Minimum Vertex Cover on triangle-free graphs of maximum degree three. Our algorithm achieves optimality and uses at most 7n/22 bits of advice. No adaptive priority algorithm without advice can achieve optimality without advice, and we prove that an online algorithm with advice needs more than 7n/22 bits of advice to reach optimality.We show connections between exact algorithms and priority algorithms with advice. The branching in branch-and-reduce algorithms can be seen as trying all possible advice strings, and all priority algorithms with advice that achieve optimality define corresponding exact algorithms, priority exact algorithms. Lower bounds on advice-based adaptive algorithms imply lower bounds on running times of exact algorithms designed in this way.

Knowledge, attitude, and practice patterns of Indian ophthalmologists regarding medicolegal issues.

To assess the knowledge, attitude, and practice patterns (KAPP) of Indian ophthalmologists regarding medicolegal issues using an initial survey. An online form was circulated among Indian ophthalmologists of all ages over social media and email by the Young Ophthalmologists Society of India (YOSI) and its medicolegal working group. Anonymous responses were obtained and analyzed for each question. The questionnaire comprised questions about demographic details, KAPP of medicolegal issues including how to deal with medicolegal issues (if faced), medical indemnity insurance, deviation from the recommended minimum sum assured (Rs. 1 crore for Indian ophthalmologists), and legal service providers. A total of 109 responses were obtained. The majority of the respondents were male (60, 55%). More than 50% of respondents (58, 53.2%) were younger than 35 years. More than one-fourth of the respondents were private practitioners (29, 27%), and the majority were of senior consultant designation (45, 41%). Around 80% of respondents (89, 81.6%) were aware of professional indemnity insurance; however, only 54% (n = 59) bought the insurance cover. A majority of the respondents (38, 64.4%) had an indemnity cover of a maximum of Rs. 50 lacs. Only 20% of respondents were aware of the expert body at the state/national level that deals with medicolegal cases. Thirty percent of respondents recommended the ideal cover amount to be more than Rs. 1 crore. The current survey highlights the dismally low rate of awareness of medicolegal issues among ophthalmologists. Specifically, a majority of Indian ophthalmologists surveyed did not have recommended minimum insured cover for professional indemnity insurance. Larger studies are needed to further explore KAPP of Indian ophthalmologists in various medicolegal issues.

On the Parallel Parameterized Complexity of MaxSAT Variants

In the maximum satisfiability problem (max-sat) we are given a propositional formula in conjunctive normal form and have to find an assignment that satisfies as many clauses as possible. We study the parallel parameterized complexity of various versions of max-sat and provide the first constant-time algorithms parameterized either by the solution size or by the allowed excess relative to some guarantee. For the dual parameterized version where the parameter is the number of clauses we are allowed to leave unsatisfied, we present the first parallel algorithm for max-2sat (known as almost-2sat). The difficulty in solving almost-2sat in parallel comes from the fact that the iterative compression method, originally developed to prove that the problem is fixed-parameter tractable at all, is inherently sequential. We observe that a graph flow whose value is a parameter can be computed in parallel and develop a parallel algorithm for the vertex cover problem parameterized above the size of a given matching. Finally, we study the parallel complexity of max-sat parameterized by the vertex cover number, the treedepth, the feedback vertex set number, and the treewidth of the input’s incidence graph. While max-sat is fixedparameter tractable for all of these parameters, we show that they allow different degrees of possible parallelization. For all four we develop dedicated parallel algorithms that are constructive, meaning that they output an optimal assignment – in contrast to results that can be obtained by parallel meta-theorems, which often only solve the decision version.

A probabilistic algorithm for vertex cover

The Minimum Vertex CoverProblem is the optimization problem of finding a vertex cover Vc of minimal cardinality in a given graph. It is a classic NP-hard problem, and various algorithms have been suggested for it.In this paper, we start with a basic algorithm for solving the problem. Using a probabilistic idea, we use it to develop an improved algorithm. The algorithm is greedy; at each step it adds to the cover a vertex such that the expected cover size, if we continue randomly after this step, is minimal. We study the new algorithm theoretically and empirically, and run simulations to compare its performance to that of some algorithms of a similar nature.

Approximating Minimum k-Tree Cover of a Connected Graph Inspired by the Multi-Ferry Routing in Delay Tolerant Networks

In this paper, we consider the problem of covering the vertex set of a given graph by [Formula: see text] trees so as to minimize the maximum weight of the trees, as a subproblem of the multi-ferry scheduling problem proposed by Zhao and Ammar. After pointing out that the approximation ratio of a greedy scheme based on the Kruskal’s algorithm is provably bad, we show that the approximation ratio cannot be better than 3/2 for [Formula: see text] even when the edge selection criterion is modified so as to minimize the increase in the maximum weight in the collection of trees. We then propose two polynomial-time algorithms with a guaranteed approximation ratio. The first algorithm achieves 3-approximation for the class of graphs in which the edge weights satisfy the triangle inequality. The second algorithm achieves 4-approximation for any connected graph with arbitrary edge weights.

Partitioning of a graph into induced subgraphs not containing prescribed cliques

Let Kp be a complete graph of order p≥2. A Kp-free k-coloring of a graph H is a partition of V(H) into V1,V2…,Vk such that H[Vi] does not contain Kp for each i≤k. In 1977 Borodin and Kostochka conjectured that any graph H with maximum degree Δ(H)≥9 and without KΔ(H) as a subgraph has chromatic number at most Δ(H)−1. As analogue of the Borodin–Kostochka conjecture, we prove that if p1≥⋯≥pk≥2, p1+p2≥7, ∑i=1kpi=Δ(H)−1+k, and H does not contain KΔ(H) as a subgraph, then there is a partition of V(H) into V1,…,Vk such that for each i, H[Vi] does not contain Kpi. In particular, if p≥4 and H does not contain KΔ(H) as a subgraph, then H admits a Kp-free ⌈Δ(H)−1p−1⌉-coloring. Catlin showed that every connected non-complete graph H with Δ(H)≥3 has a Δ(H)-coloring such that one of the color classes is maximum K2-free subset (maximum independent set). In this regard, we show that there is a partition of vertices of H into V1 and V2 such that V1 is a maximum Kp-free subset of H and V2 is a Kq-free subset of H, if p≥4, q≥3, p+q=Δ(H)+1, and its clique number ω(H)=p.

Modular polymer stormwater collection structure response to one-week design-truck load

Full-scale physical modelling was used to evaluate the response of modular polymer stormwater collection structures when subjected to design-truck loading that was sustained for 1 week (i.e. a parked vehicle) and when buried with minimum soil cover. Surface loading was applied on a steel load pad that simulates one-half of the AASHTO design-truck single-axle that was held constant for 1 week to assess the stability of the buried system and demand on the platen and column components of the modules to allow independent load rating of the buried structure. No collapse or limiting state occurred. The 1-week creep buckling resistance was found to be nearly four times the factored column demand to 1-week loading for both module types. New insight into the time-dependent behaviour of the soil–structure system was gained by comparing the buried response to creep tests conducted on isolated columns and platens. The response of the buried soil–structure system was less severe than unconstrained creep. Column loads were found to not increase with time. Bending deflections increased by 1.04 to 1.26 times when buried rather than by 3.0 to 3.4 times for unconstrained creep as platen creep when buried was greatly restrained by the soil.

Disentangling the Computational Complexity of Network Untangling

We study the network untangling problem introduced by Rozenshtein et al. (Data Min. Knowl. Disc. 35(1), 213–247, 2021), which is a variant of Vertex Cover on temporal graphs–graphs whose edge set changes over discrete time steps. They introduce two problem variants. The goal is to select at most k time intervals for each vertex such that all time-edges are covered and (depending on the problem variant) either the maximum interval length or the total sum of interval lengths is minimized. This problem has data mining applications in finding activity timelines that explain the interactions of entities in complex networks. Both variants of the problem are NP-hard. In this paper, we initiate a multivariate complexity analysis involving the following parameters: number of vertices, lifetime of the temporal graph, number of intervals per vertex, and the interval length bound. For both problem versions, we (almost) completely settle the parameterized complexity for all combinations of those four parameters, thereby delineating the border of fixed-parameter tractability.

On the complexity of robust multi-stage problems with discrete recourse

We study the computational complexity of multi-stage robust optimization problems. Such problems are formulated with alternating min/max quantifiers and therefore naturally fall into a higher stage of the polynomial hierarchy. Despite this, almost no hardness results with respect to the polynomial hierarchy are known.In this work, we examine the hardness of robust two-stage adjustable and robust recoverable optimization with budgeted uncertainty sets. Our main technical contribution is the introduction of a technique tailored to prove Σ3p-hardness of such problems. We highlight a difference between continuous and discrete budgeted uncertainty: In the discrete case, indeed a wide range of problems becomes complete for the third stage of the polynomial hierarchy; in particular, this applies to the TSP, independent set, and vertex cover problems. However, in the continuous case this does not happen and problems remain in the first stage of the hierarchy. Finally, if we allow the uncertainty to not only affect the objective, but also multiple constraints, then this distinction disappears and even in the continuous case we encounter hardness for the third stage of the hierarchy. This shows that even robust problems which are already NP-complete can still exhibit a significant difference in computational complexity between column-wise and row-wise uncertainty.

Green conditionality and eco-schemes in the CAP Strategic Plan of Bulgaria

Bulgaria's CAP Strategic Plan for developing agriculture and rural areas for 2023 – 2027 is implemented for the first year by farmers, with the campaign for submitting support applications ending in July 2023. The purpose of the present study is to investigate whether Bulgarian grain producers encounter challenges regarding the compliance and implementation of the "green architecture" within the framework of direct payments under the Strategic Plan. In this regard, the possibilities of the grain producers in terms of compliance with the mandatory standards for the good agricultural and ecological condition of the land, as well as the farmers’ attitude to participate in one-year voluntary eco-schemes, were investigated. The survey results show that the most significant difficulties for grain producers are registered in compliance with standards for minimum soil cover to avoid bare soil in periods and areas that are most sensitive and for a minimum share of arable land devoted to non-productive areas and features. Surprisingly, the survey shows significant interest in grain producers' participation in an eco-scheme to reduce the use of pesticides. The worrying results of the research are that amongst the country's producers of grain and oil crops, farmers who did not apply for support under the basic scheme have been identified. Also, part of the grain producers would refuse to participate in support under the European funds because of the increased environmental requirements. Moreover, cereal producers who would give up agricultural activity if the environmental requirements for farmers are further increased have also been identified. Keywords: CAP Strategic Plan, green architecture, green conditionality, GAECs, eco-schemes, grain producers

Stability, Vertex Stability, and Unfrozenness for Special Graph Classes

Frei et al. (J. Comput. Syst. Sci. 123, 103–121, 2022) show that the stability, vertex stability, and unfrozenness problems with respect to certain graph parameters are complete for Θ2P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\\Theta _{2}^{\ extrm{P}}}$$\\end{document}, the class of problems solvable in polynomial time by parallel access to an NP oracle. They studied the common graph parameters α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\\alpha }$$\\end{document} (the independence number), β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\\beta }$$\\end{document} (the vertex cover number), ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\\omega }$$\\end{document} (the clique number), and χ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\\chi }$$\\end{document} (the chromatic number). We complement their approach by providing polynomial-time algorithms solving these problems for special graph classes, namely for graphs with bounded tree-width or bounded clique-width. In order to improve these general time bounds even further, we then focus on trees, forests, bipartite graphs, and co-graphs.

APPROXIMATION ALGORITHMS WITH CONSTANT FACTORS FOR A SERIES OF ASYMMETRIC ROUTING PROBLEMS

In this paper, the first fixed-ratio approximation algorithms are proposed for the series of asymmetric settings of the well-known combinatorial routing problems. Among them are the Steiner cycle problem, the prize-collecting traveling salesman problem, the minimum cost cycle cover problem by a bounded number of cycles, etc. Almost all the proposed algorithms rely on original reductions of the considered problems to auxiliary instances of the Asymmetric Traveling Salesman Problem and employ the breakthrough approximation results for this problem obtained recently by O. Svensson, J. Tarnawski, L. Végh, V. Traub and J. Vygen. On the other hand, approximation of the cycle cover problem was proved by more deep extension of their approach.

Sharp Bounds and Precise Values for the Ni-Chromatic Number of Graphs

Let [Formula: see text] be a connected undirected graph. A vertex coloring [Formula: see text] of [Formula: see text] is an [Formula: see text]-vertex coloring if for each vertex [Formula: see text] in [Formula: see text], the number of different colors assigned to [Formula: see text] is at most [Formula: see text]. The [Formula: see text]-chromatic number of [Formula: see text], denoted by [Formula: see text], is the maximum number of colors which are used in an [Formula: see text]-vertex coloring of [Formula: see text]. In this paper, we provide sharp bounds for [Formula: see text] of a graph [Formula: see text] in terms of its vertex cover number, maximum degree and diameter, respectively. We also determine precise values for [Formula: see text] in some cases.

The covering numbers of rings

A cover of an associative (not necessarily commutative nor unital) ring R is a collection of proper subrings of R whose set-theoretic union equals R. If such a cover exists, then the covering number σ(R) of R is the cardinality of a minimal cover, and a ring R is called σ-elementary if σ(R)<σ(R/I) for every nonzero two-sided ideal I of R. If R is a ring with unity, then we define the unital covering number σu(R) to be the size of a minimal cover of R by subrings that contain 1R (if such a cover exists), and R is σu-elementary if σu(R)<σu(R/I) for every nonzero two-sided ideal of R. In this paper, we classify all σ-elementary unital rings and determine their covering numbers. Building on this classification, we are further able to classify all σu-elementary rings and prove σu(R)=σ(R) for every σu-elementary ring R. We also prove that, if R is a ring without unity with a finite cover, then there exists a unital ring R′ such that σ(R)=σu(R′), which in turn provides a complete list of all integers that are the covering number of a ring. Moreover, ifE(N):={m:m⩽N,σ(R)=m for some ring R}, then we show that |E(N)|=Θ(N/log⁡N), which proves that almost all integers are not covering numbers of a ring.

Large Independent Sets on Random d-Regular Graphs with Fixed Degree d

The maximum independent set problem is a classic and fundamental combinatorial challenge, where the objective is to find the largest subset of vertices in a graph such that no two vertices are adjacent. In this paper, we introduce a novel linear prioritized local algorithm tailored to address this problem on random d-regular graphs with a small and fixed degree d. Through exhaustive numerical simulations, we empirically investigated the independence ratio, i.e., the ratio between the cardinality of the independent set found and the order of the graph, which was achieved by our algorithm across random d-regular graphs with degree d ranging from 5 to 100. Remarkably, for every d within this range, our results surpassed the existing lower bounds determined by theoretical methods. Consequently, our findings suggest new conjectured lower bounds for the MIS problem on such graph structures. This finding has been obtained using a prioritized local algorithm. This algorithm is termed ‘prioritized’ because it strategically assigns priority in vertex selection, thereby iteratively adding them to the independent set.