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

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

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

The generalized vertex cover problem and some variations

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 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 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).

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.

A rough set method for the minimum vertex cover problem of graphs

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.

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.

Parameterized approximation algorithms for weighted vertex cover

The parameterized complexity of finding secluded solutions to some classical optimization problems on graphs

In this paper, we propose to study a wide class of combinatorial optimization problems called combinatorial optimization problems with labelling. First, we give a combinatorial method to deal with the labelling version of some classical combinatorial optimization problems including minimum vertex cover, maximum independent set, minimum dominating set and minimum set cover, and convert the labelling problems into the original problems by polynomial-time reduction. We show that, although the labelling version of these problem seems more universal than their original counterparts, they are actually equivalent to the corresponding original problem from an algorithmic point of view. Moreover, we generalize the greedy approach for solving submodular cover problem to its labelling version, and as simple applications of our new method, we use it to solve the labelling versions of the minimum weighted spanning tree and connected vertex cover problem in a unified way.

We study the Minimum Sum Vertex Cover problem, which asks for an ordering of vertices in a graph that minimizes the total cover time of edges. In particular, n vertices of the graph are visited according to an ordering, and for each edge this induces the first time it is covered. The goal of the problem is to find the ordering that minimizes the sum of the cover times over all edges in the graph. In this work, we give the first explicit hardness of approximation result for Minimum Sum Vertex Cover. In particular, assuming the Unique Games Conjecture, we show that the Minimum Sum Vertex Cover problem cannot be approximated within 1.0748. The best approximation ratio for Minimum Sum Vertex Cover as of now is 16/9, due to a recent work by Bansal, Batra, Farhadi, and Tetali. We also study the Minimum Sum Vertex Cover problem on regular graphs. In particular, we show that in this case the problem is hard to approximate within 1.0157. We also revisit an approximation algorithm for regular graphs outlined in the work of Feige, Lovász, and Tetali to show that Minimum Sum Vertex Cover can be approximated within 1.225 on regular graphs.

The following planar minimum disk cover problem is considered in this paper: given a set $\mathcal{D}$ of n disks and a set ? of m points in the Euclidean plane, where each disk covers a subset of points in ?, to compute a subset of disks with minimum cardinality covering ?. This problem is known to be NP-hard and an algorithm which approximates the optimal disk cover within a factor of (1+?) in $\mathcal{O}(mn^{\mathcal{O}(\frac{1}{\epsilon^{2}}\log^{2}\frac{1}{\epsilon})})$ time is proposed in this paper. This work presents the first polynomial time approximation scheme for the minimum disk cover problem where the best known algorithm can approximate the optimal solution with a large constant factor. Further, several variants of the minimum disk cover problem such as the incongruent disk cover problem and the weighted disk cover problem are considered and approximation schemes are designed.