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

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

The minimum vertex cover (MVC) problem is a classic NP-hard combinatorial optimization problem with extensive real-world applications. In this paper, we propose an efficient local search algorithm, InfVC, to solve the MVC in massive graphs, which comprises three ideas. First, we introduce an inference-driven optimization strategy that explores better feasible solutions through inference rules. Second, we develop a structural-determined perturbation strategy that is motivated by the structure features of high-quality solutions, prioritizing high-degree vertices into the candidate solution to guide the search process to some potential high-quality search area. Third, we design a self-adaptive local search framework that dynamically balances exploration and exploitation through a perturbation management mechanism. Extensive experiments demonstrate that InfVC outperforms all the state-of-the-art algorithms on almost massive instances.

Minimum weighted vertex cover (MWVC) problem of weighted networks is one of the most important combinatorial optimization problems in network optimization. This paper models the MWVC problem as an asymmetric game on weighted networks. Then, an asynchronous game-based algorithm (AGB) is proposed to solve the MWVC problem, which can guarantee to obtain a vertex cover (VC). Numerical illustrations verify the performance of the proposed game solution on the representative weighted networks.

The generalized vertex cover problem, an extension of classic minimum vertex cover problem, is an important NP-hard combinatorial optimization problem with a wide range of applications. The aim of this paper is to design an efficient local search algorithm with tabu strategy and perturbation mechanism to solve this problem. Firstly, we use tabu strategy to prevent the local search from immediately returning to a previously visited candidate solution and avoiding the cycling problem. Secondly, we propose the flip gain for each vertex, and then the tabu strategy is combined with the flip gain for vertex selecting. Finally, we apply a simple perturbation mechanism to help the search to escape from deep local optima and to bring diversification into the search. The experiments are carried on random instances with up to 1000 vertexes and 450,000 edges. The experimental results show that our algorithm performs better than a state-of-art algorithm in terms of both solution quality and computational efficiency in most instances.

The Weighted Constraint Satisfaction Problem (WCSP) is a very expressive framework for optimization problems. The Constraint Composite Graph (CCG) is a graphical representation of a given (Boolean) WCSP that facilitates its reduction to a Minimum Weighted Vertex Cover (MWVC) problem by introducing intelligently chosen auxiliary variables. It also enables kernelization: a maxflow procedure used to fix the optimal values of a subset of the variables before initiating search. In this paper, we present some CCG-based WCSP solvers and compare their performance against toulbar2, a state-of-the-art WCSP solver, on a variety of benchmark instances. We also study the effectiveness of kernelization.

We introduce two generalizations of core percolation in graphs to hypergraphs, related to the minimum hyperedge cover problem and the minimum vertex cover problem on hypergraphs, respectively. We offer analytical solutions of these two core percolations for uncorrelated random hypergraphs whose vertex degree and hyperedge cardinality distributions are arbitrary but have nondiverging moments. We find that for several real-world hypergraphs their two cores tend to be much smaller than those of their null models, suggesting that covering problems in those real-world hypergraphs can actually be solved in polynomial time.

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.

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...

The generalized vertex cover problem and some variations

The minimum vertex cover (MVC) problem is a well-studied NP-Complete problem and has various applications. In this paper, a new heuristic approach has been proposed to find the minimum vertex cover of a graph. The proposed algorithm has been tested on random graphs and BHOSLIB instances. The results have shown that the proposed algorithm can yield better solutions especially on dense graphs for solving the minimum vertex cover problem.

We present a deterministic distributed 2-approximation algorithm for the Minimum Weight Vertex Cover problem in the CONGEST model whose round complexity is \(O(\log n\log \varDelta / \log ^2\log \varDelta )\). This improves over the currently best known deterministic 2-approximation implied by [KVY94]. Our solution generalizes the \((2+\epsilon )\)-approximation algorithm of [BCS17], improving the dependency on \(\epsilon ^{-1}\) from linear to logarithmic. In addition, for every \(\epsilon =(\log \varDelta )^{-c}\), where \(c\ge 1\) is a constant, our algorithm computes a \(\left( 2+\epsilon \right) \)-approximation in \(O(\log {\varDelta }/\log \log {\varDelta })\) rounds (which is asymptotically optimal).

Geometric Covering and Packing problems have been extensively studied in the last few decades and have applications in diverse areas. Several variants and generalizations of these problems have been studied recently. In this paper, we look at the following covering variants where we require that each point is “uniquely” covered, i.e., it is covered by exactly one object: Unique Coverage problem, where we want to maximize the number of uniquely covered points and Exact Cover problem, where we want to uniquely cover every point and minimize the number of objects used for covering. We also look at the following generalizations: Multi Cover problem, a generalization of Set Cover, the objective is to select the minimum subset of objects with the constraint that each input point p is covered by at least $$d_p$$ objects in the solution, where $$d_p$$ is the demand of point p. And Shallow Packing problem, a generalization of Packing problem, where we want to select the maximum subset of objects with the constraint that any point in the plane is contained in at most k objects in the solution. The above problems are NP-hard even for unit squares in the plane. Thus, the focus has been on obtaining good approximation algorithms. Local search has been quite successful in the recent past in obtaining good approximation algorithms for a wide variety of problems. We consider the Unique Coverage and Multi Cover problems on non-piercing objects, which is a broad class that includes squares, disks, pseudo-disks, etc. and show that the local search algorithm yields a PTAS approximation under the assumption that the depth of every input point is at most some fixed constant. For Unique Coverage we further assume that every object has at most a constant degree. For the Shallow Packing problem, we show that the local search algorithm yields a PTAS approximation for objects with sub-quadratic union complexity, which is a very broad class of objects that even includes non-piercing objects. For the Exact Cover problem, we show that finding a feasible solution is NP-hard even for unit squares in the plane, thus negating the existence of polynomial time approximation algorithms.

To introduce our question and the parameterization, consider the classical Vertex Cover problem. In this problem, the input is a graph G on n vertices and a positive integer \(\ell \), and the goal is to find a vertex subset S of size at most \(\ell \) such that \(G-S\) is an independent set. Further, we want that G[S] is highly connected. That is, G[S] should be \(n-k\) edge-connected. Clearly, the problem is NP-complete, as substituting \(k=n-1\), we obtain the Connected Vertex Cover problem. A simple observation also shows that the problem admits an algorithm with running time \(n^{{\mathcal O}(k)}\). Since the problem is polynomial-time solvable for every fixed integer k, a natural parameter is the integer k. In all the problems we consider, the parameter is k, and the goal is to find a solution S of size at most \(\ell \), such that G[S] is \(n-k\) edge-connected and \(G-S\) satisfies a property. We show that this version of well-known problems such as Vertex Cover, Feedback Vertex Set, Odd Cycle Transversal and Multiway Cut admit an algorithm with running time \(f(k)\cdot n^{{\mathcal O}(1)}\), that is, they are FPT with the parameter k. One of our main subroutines to obtain these algorithms is an FPT algorithm for \(n-k\) edge connected Steiner Subgraph, which could be of an independent interest. Finally, we also show that such an algorithm is not possible for Multicut.Keywordssubset problemsparameterized algorithmsconnectivity

Parameterized algorithms for finding highly connected solution

The minimum sensor cover is an important optimization problem in wireless sensor networks, which can be seen as a special case of the minimum set cover problem. The minimum set cover problem is a well-known problem in combinatorial optimization, which has no polynomial-time (ρlnδ)-approximation for 0<ρ<1 unless NP⊆DTIME(n O(loglogn)) where δ is the maximum cardinality of a subset in the input collection. However, the minimum sensor cover problem has polynomial-time constant-approximations because this special case has a geometric structure. The design technique, called partition, is employed to take the advantage of this geometric structure. Especially, the double partition plays an important role. In this article, we would like to give an exploratory essay for the double partition technique together with research progress on approximations for the minimum sensor cover problem.