Minimum Weight Vertex Cover Problem Research Articles

Ising Hamiltonians for Constrained Combinatorial Optimization Problems and the Metropolis‐Hastings Warm‐Starting Algorithm

AbstractQuantum approximate optimization algorithm (QAOA) is a promising variational quantum algorithm for combinatorial optimization problems. However, the implementation of QAOA is limited due to the requirement that the problems be mapped to Ising Hamiltonians and the nonconvex optimization landscapes. Although the Ising Hamiltonians for many NP hard problems have been obtained, a general method to obtain the Ising Hamiltonians for constrained combinatorial optimization problems (CCOPs) has not yet been investigated. In this paper, a general method is introduced to obtain the Ising Hamiltonians for CCOPs and the Metropolis‐Hastings warm‐starting algorithm for QAOA is presented which can provably converge to the global optimal solutions. The effectiveness of this method is demonstrated by tackling the minimum weight vertex cover (MWVC) problem, the minimum vertex cover (MVC) problem, and the maximal independent set problem as examples. The Ising Hamiltonian for the MWVC problem is obtained first time by using this method. The advantages of the Metropolis‐Hastings warm‐starting algorithm presented here is numerically analyzed through solving 30 randomly generated MVC cases with 1‐depth QAOA.

An evolutionary game algorithm for minimum weighted vertex cover problem

An evolutionary game algorithm for minimum weighted vertex cover problem

Better Approximation for Distributed Weighted Vertex Cover via Game-Theoretic Learning

Toward better approximation for the minimum-weighted vertex cover (MWVC) problem in multiagent systems, we present a distributed algorithm from the perspective of learning in games. For self-organized coordination and optimization, we see each vertex as a potential game player who makes decisions using local information of its own and the immediate neighbors. The resulting Nash equilibrium is classified into two categories, i.e., the inferior Nash equilibrium (INE) and the dominant Nash equilibrium (DNE). We show that the optimal solution must be a DNE. To achieve better approximation ratios, local rules of perturbation and weighted memory are designed, with the former destroying the stability of an INE and the latter facilitating the refinement of a DNE. By showing the existence of an improvement path from any INE to a DNE, we prove that when the memory length is larger than 1, our algorithm converges in finite time to DNEs, which could not be improved by exchanging the action of a selected node with all its unselected neighbors. Moreover, additional freedom for solution efficiency refinement is provided by increasing the memory length. Finally, intensive comparison experiments demonstrate the superiority of the presented methodology to the state of the art, both in solution efficiency and computation speed.

Fixed set search applied to the multi-objective minimum weighted vertex cover problem

The Fixed Set Search (FSS) is a novel metaheuristic that adds a learning mechanism to the Greedy Randomized Adaptive Search Procedure (GRASP). In recent publications, its efficiency has been shown on different types of combinatorial optimization problems like routing, machine scheduling and covering. In this paper the FSS is adapted to multi-objective problems for finding Pareto Front approximations. This adaptation is illustrated for the bi-objective Minimum Weighted Vertex Cover Problem (MWVCP). In this work, a simple and effective bi-objective GRASP algorithm for the MWVCP is developed in the first stage. One important characteristic of the proposed GRASP is that it avoids the use of weighted sums of objective functions in the local search and the greedy algorithm. In the second stage, the bi-objective GRASP is extended to the FSS by adding a learning mechanism adapted to multi-objective problems. The conducted computational experiments show that the proposed FSS and GRASP algorithm significantly outperforms existing methods for the bi-objective MWVCP. To fully evaluate the learning mechanism of the FSS, it is compared to the underlying GRASP algorithm on a wide range of performance indicators related to convergence, distribution, spread and cardinality.

A Fast and Robust Heuristic Algorithm for the Minimum Weight Vertex Cover Problem

The minimum weight vertex cover problem (MWVCP) is a fundamental combinatorial optimization problem with various real-world applications. The MWVCP seeks a vertex cover of an undirected graph such that the sum of the weights of the selected vertices is as small as possible. In this paper, we present an effective algorithm to solve the MWVCP. First, a master-apprentice evolutionary algorithm based on two individuals is conducted to enhance the diversity of solutions. Second, a hybrid tabu search combined configuration checking and solution-based tabu search is introduced to intensify local search procedure. Harnessing the power of the evolutionary strategy and a novel variant of hybrid tabu search, Master-Apprentice Evolutionary Algorithm with Hybrid Tabu Search, MAE-HTS, is presented. Results of extensive computational experiments using standard benchmark instances and other large-scale instances demonstrate the efficacy of our algorithm in terms of solution quality, running time, and robustness compared to state-of-the-art heuristics from the literature and the commercial MIP solver Gurobi. We also systematically analyze the role of each individual component of the algorithm which when worked in unison produced superior outcomes. In particular, MAE-HTS produced improved solutions for 2 out of 126 public benchmark instances with better running time. In addition, our MAE-HTS outperforms other state-of-the-art algorithms DLSWCC and NuMWVC for 72 large scale MWVCP instances by obtaining the best results for 64 ones, while other two reference algorithms can only obtain 27 best results at most.

Jumping particle swarm optimisation method for solving minimum weight vertex cover problem

The minimum weight vertex cover problem (MWVCP) is one of the classic combinatorial optimisation problems. In this paper, we have proposed a modified jumping particle swarm optimisation approach, n...

Jumping particle swarm optimisation method for solving minimum weight vertex cover problem

Jumping particle swarm optimisation method for solving minimum weight vertex cover problem

Optimized Energy – Efficient Path Planning Strategy in WSN With Multiple Mobile Sinks

Recently, wireless sensor networks (WSNs) were perceived as the foundation infrastructure that paved the way to the emergence of the Internet of Things (IoT). However, a challenging issue exists when WSNs are integrated into the IoT because of high energy consumption in their nodes and poor network lifespan. Therefore, the elementary discussions in WSN are energy scarcity in sensor nodes, sensors' data exchange, and routing protocols. To address the aforesaid shortcomings, this paper develops an optimized energy-efficient path planning strategy that prolongs the network lifetime and enhances its connectivity. The proposed approach has four successive procedures: initially, the sensing field is partitioned into equal regions depending on the number of deployed mobile sinks that eliminate the energy-hole problem. A new heuristic clustering approach called stable election algorithm (SEA) is introduced to minimize the message exchange between sensor nodes and prevent frequent cluster heads rotation. A sojourn location determination algorithm is proposed based on the minimum weighted vertex cover problem (MWVCP) to find the best position for the sinks to stop and collect the data from cluster heads. Finally, three optimization techniques are utilized to evaluate the optimized mobile sinks' trajectories using multi-objective evolutionary algorithms (MOEAs). Whilst the performance of the developed work was evaluated in terms of cluster heads number, network lifetime, the execution time of the sinks' sojourn locations determination algorithm, the convergence rate of optimization techniques, and data delivery. The simulation scenarios conducted in MATLAB and the obtained results showed that the introduced approach outperformed comparable existing schemes. It succeeded in prolonging the network lifetime up to 66% compared to existing routing protocols.

Risk-Aware Identification of Highly Suspected COVID-19 Cases in Social IoT: A Joint Graph Theory and Reinforcement Learning Approach.

The recent outbreak of the coronavirus disease 2019 (COVID-19) has rapidly become a pandemic, which calls for prompt action in identifying suspected cases at an early stage through risk prediction. To suppress its further spread, we exploit the social relationships between mobile devices in the Social Internet of Things (SIoT) to help control its propagation by allocating the limited protective resources to the influential so-called high-degree individuals to stem the tide of precipitated spreading. By exploiting the so-called differential contact intensity and the infectious rate in susceptible-exposed-infected-removed (SEIR) epidemic model, the resultant optimization problem can be transformed into the minimum weight vertex cover (MWVC) problem of graph theory. To solve this problem in a high-dynamic random network topology, we propose an adaptive scheme by relying on the graph embedding technique during the state representation and reinforcement learning in the training phase. By relying on a pair of real-life datasets, the results demonstrate that our scheme can beneficially reduce the epidemiological reproduction rate of the infection. This technique has the potential of assisting in the early identification of COVID-19 cases.

Multi-Objective Neighborhood Search Algorithm Based on Decomposition for Multi-Objective Minimum Weighted Vertex Cover Problem

The multi-objective minimum weighted vertex cover problem aims to minimize the sum of different single type weights simultaneously. In this paper, we focus on the bi-objective minimum weighted vertex cover and propose a multi-objective algorithm integrating iterated neighborhood search with decomposition technique to solve this problem. Initially, we adopt the decomposition method to divide the multi-objective problem into several scalar optimization sub-problems. Meanwhile, to find more possible optimal solutions, we design a mixed score function according to the problem feature, which is applied in initializing procedure and neighborhood search. During the neighborhood search, three operators ( A d d , D e l e t e , S w a p ) explore the search space effectively. We performed numerical experiments on many instances, and the results show the effectiveness of our new algorithm (combining decomposition and neighborhood search with mixed score) on several experimental metrics. We compared our experimental results with the classical multi-objective algorithm non-dominated sorting genetic algorithm II. It was obviously shown that our algorithm can provide much better results than the comparative algorithm considering the different metrics.

NuMWVC: A novel local search for minimum weighted vertex cover problem

The problem of finding a minimum weighted vertex cover (MWVC) in a graph is a well-known combinatorial optimisation problem with important applications. This article introduces a novel local search algorithm called NuMWVC for MWVC based on three ideas. First, four reduction rules are introduced during the initial construction phase. Second, a strategy called configuration checking with aspiration, which aims for reducing cycling in local search, is proposed for MWVC for the first time. Moreover, a self-adaptive vertex removing strategy is proposed to save time spent on searching solutions for which the quality is likely far from optimality. Experimental results show that NuMWVC outperforms state-of-the-art local search algorithms for MWVC on the standard benchmarks, massive graphs and real-world problem (map labeling problem) instances.

Towards faster local search for minimum weight vertex cover on massive graphs

The minimum weight vertex cover (MWVC) problem is a well known NP-hard problem with various real-world applications. In this paper, we design an efficient algorithm named FastWVC to solve MWVC problem in massive graphs. To this end, we propose a construction procedure, which aims to generate a quality initial vertex cover in short time. We also propose a new exchange step for reconstructing a vertex cover. Additionally, a cost-effective strategy is used for choosing adding vertices, which can accelerate the algorithm. Experiments on 102 instances were conducted to confirm the effectiveness of our algorithm. The results show that the FastWVC algorithm outperforms other algorithms in terms of both solution quality and computational time in most of the instances. We also carry out experiments that analyze the effectiveness of the underlying ideas.

The generalized vertex cover problem and some variations

In this paper we study the generalized vertex cover problem (GVC), which is a generalization of various well studied combinatorial optimization problems. GVC is shown to be equivalent to the unconstrained binary quadratic programming problem and also equivalent to some other variations of the general GVC. Some solvable cases are identified and approximation algorithms are suggested for special cases. We also study GVC on bipartite graphs and identify some polynomially solvable cases. We show that GVC on bipartite graphs is equivalent to the bipartite unconstrained 0-1 quadratic programming problem. Integer programming formulations of GVC and related problems are presented and establish half-integrality property on some variables for the corresponding linear programming relaxations. We also discuss special cases of GVC where all feasible solutions are independent sets or vertex covers. These problems are observed to be equivalent to the maximum weight independent set problem or minimum weight vertex cover problem along with some algorithmic results.

NuMWVC: A Novel Local Search for Minimum Weighted Vertex Cover Problem

The minimum weighted vertex cover (MWVC) problem is a well known combinatorial optimization problem with important applications. This paper introduces a novel local search algorithm called NuMWVC for MWVC based on three ideas. First, four reduction rules are introduced during the initial construction phase. Second, the configuration checking with aspiration is proposed to reduce cycling problem. Moreover, a self-adaptive vertex removing strategy is proposed to save time.

Test-cost-sensitive rough set based approach for minimum weight vertex cover problem

The minimum vertex cover problem (MVCP) and minimum weighted vertex cover problem (MWVCP) have been used in a variety of applications. This paper focuses on a view of test-cost-sensitive rough set for MWVCP. We first provide a method to convert a minimum weight vertex cover of a graph into a minimal test cost attribute reduct of a test-cost-sensitive decision table. Then, an induced test-cost-sensitive decision table from an undirected weighted graph is established. On the foundation of the induced decision table, an improved heuristic algorithm for finding minimum weight vertex covers is proposed, it can avoid a mass of redundant computation. Furthermore, to improve efficiency, a quantum-behaved particle swarm optimization with immune mechanism is presented, which can avoid the phenomenon of premature, improve the global searching ability, and enhance the convergence speed. The results of the experiment show the advantages and limitations of the proposed algorithms compared with state-of-the-art algorithms.

Carousel greedy: A generalized greedy algorithm with applications in optimization

In this paper, we introduce carousel greedy, an enhanced greedy algorithm which seeks to overcome the traditional weaknesses of greedy approaches. We have applied carousel greedy to a variety of well-known problems in combinatorial optimization such as the minimum label spanning tree problem, the minimum vertex cover problem, the maximum independent set problem, and the minimum weight vertex cover problem. In all cases, the results are very promising. Since carousel greedy is very fast, it can be used to solve very large problems. In addition, it can be combined with other approaches to create a powerful, new metaheuristic. Our goal in this paper is to motivate and explain the new approach and present extensive computational results.

An efficient local search framework for the minimum weighted vertex cover problem

The minimum weighted vertex cover (MWVC) problem, an extension of the classical minimum vertex cover (MVC) problem, is an important NP-complete combinatorial optimization problem with a wide range of applications. The objective of this paper is to design an efficient local search algorithm to solve the MWVC problem. First, the weighted edge strategy is proposed to define the dynamic scoring strategy so that our algorithm can find different possible optimal solutions. Second, the weighted configuration checking (WCC) strategy is proposed to overcome the cycling problem in local search. By combining the WCC strategy with the scoring strategy, we design the vertex selection strategy to determine the vertex to be selected as a candidate solution component. Based on these strategies, a novel local search framework, namely diversion local search based on weighted configuration checking (DLSWCC), is presented. DLSWCC is evaluated against several state-of-the-art algorithms on various benchmark instances. Experimental results show that DLSWCC outperforms its competitors in terms of both solution quality and computational efficiency in most classical instances. Specifically, DLSWCC can obtain 22 new upper bounds of 71 moderate-scale problem instances, 5 new upper bounds of 15 large-scale problem instances, and 56 new upper bounds of 56 massive graph instances.

Multi-start iterated tabu search for the minimum weight vertex cover problem

The minimum weight vertex cover problem (MWVCP) is one of the most popular combinatorial optimization problems with various real-world applications. Given an undirected graph where each vertex is w...

Fair Matchings and Related Problems

Let \(G = (A \cup B, E)\) be a bipartite graph, where every vertex ranks its neighbors in an order of preference (with ties allowed) and let \(r\) be the worst rank used. A matching \(M\) is fair in \(G\) if it has maximum cardinality, subject to this, \(M\) matches the minimum number of vertices to rank \(r\) neighbors, subject to that, \(M\) matches the minimum number of vertices to rank \((r-1)\) neighbors, and so on. We show an efficient combinatorial algorithm based on LP duality to compute a fair matching in \(G\). We also show a scaling based algorithm for the fair b-matching problem. Our two algorithms can be extended to solve other profile-based matching problems. In designing our combinatorial algorithm, we show how to solve a generalized version of the minimum weighted vertex cover problem in bipartite graphs, using a single-source shortest paths computation—this can be of independent interest.

Approximation algorithm for the minimum weight connected k-subgraph cover problem

A subset F of vertices is called a connected k-subgraph cover (VCCk) if every connected subgraph on k vertices contains at least one vertex from F. The minimum weight connected k-subgraph cover problem (MWVCCk) has its background in the field of security and supervisory control. It is a generalization of the minimum weight vertex cover problem, and is related with the minimum weight k-path cover problem (MWVCPk) which requires that every path on k vertices has at least one vertex from F. A k-approximation algorithm can be easily obtained by LP rounding method. Assuming that the girth of the graph is at least k, we reduce the approximation ratio to k−1, which is tight for our algorithm.