An Asynchronous Game-based algorithm to the Weighted Vertex Cover of Networks

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 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 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.

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 generalized vertex cover problem and some variations

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).

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

We present a time-optimal deterministic distributed algorithm for approximating a minimum weight vertex cover in hypergraphs of rank f. This problem is equivalent to the Minimum Weight Set Cover problem in which the frequency of every element is bounded by f. The approximation factor of our algorithm is (f+varepsilon ). Let varDelta denote the maximum degree in the hypergraph. Our algorithm runs in the congest model and requires O(log {varDelta } / log log varDelta ) rounds, for constants varepsilon in (0,1] and fin {mathbb {N}}^+. This is the first distributed algorithm for this problem whose running time does not depend on the vertex weights nor the number of vertices. Thus adding another member to the exclusive family of provably optimal distributed algorithms. For constant values of f and varepsilon , our algorithm improves over the (f+varepsilon )-approximation algorithm of Kuhn et al. (SODA, 2006)whose running time is O(log varDelta + log W), where W is the ratio between the largest and smallest vertex weights in the graph. Our algorithm also achieves an f-approximation for the problem in O(flog n) rounds, improving over the classical result of Khuller et al. (J Algorithms, 1994) that achieves a running time of O(flog ^2 n). Finally, for weighted vertex cover (f=2) our algorithm achieves a deterministic running time of O(log n), matching the randomized previously best result of Koufogiannakis and Young (Distrib Comput, 2011). We also show that integer covering-programs can be reduced to the Minimum Weight Set Cover problem in the distributed setting. This allows us to achieve an (flceil log _2(M)+1 rceil +varepsilon )-approximate integral solution in O(1+f/logn)·logΔloglogΔ+(f·logM)1.01·logε-1·(logΔ)0.01\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} O\\left( (1+f/\\log n)\\cdot \\left( {\\frac{\\log \\varDelta }{ \\log \\log \\varDelta } + ({f\\cdot \\log M})^{1.01}\\cdot \\log \\varepsilon ^{-1}\\cdot (\\log \\varDelta )^{0.01}}\\right) \\right) \\end{aligned}$$\\end{document}rounds, where f bounds the number of variables in a constraint, varDelta bounds the number of constraints a variable appears in, and M=max left{ 1, lceil 1/a_{min } rceil right} , where a_{min } is the smallest normalized constraint coefficient.

Chapter 1 Applications of network optimization

Weighted vertex cover (WVC) is a classical combinatorial optimization problem, which has rich practice significance. Indeed, the existing WVC studies based on game theory has introduced the traditional expected utility theory (EUT), where each vertex is treated as a rational player. However, the uncertainty caused by the weight factor is neglected. So how does this uncertainty affect players' behaviour? In this paper, the complete theoretical analysis is shown. Specifically, we model the WVC problem as an non-cooperative game on weighted networks and respectively establish EUT-based and prospect theory-based utility functions. Then, we prove the existence of mixed NE on this weighted game. Further, by designing a state function, we find that the vertices' state under mixed NE satisfies weighted vertex cover state of the network. Finally, a fictitious play algorithm is proposed to get a NE of the game. Our findings will pave the way for the future numerical simulation, which can demonstrate the impact of the prospect theory on weighted vertex cover.

Mathematical models do not exist in vacuum, but are typically representations of real-life problem situations. Therefore, even in a book of problems in network optimization, it is important to point out the position of optimization modeling and solution in real life problem solving. Hence this chapter.Modeling and solving a practical problem is usually a long process of advancing insight. The person on the work floor, who actually faces the problem, often intuitively feels the presence of a problem, and gradually, in a step-by-step process, identifies the actual problem. Only then it is time to think of solving the problem, and possibly implementing a computer decision support system to help the organization more accurate and effective in the future when similar problems arise. Very often organizations do not take enough time to analyze the problem in enough detail and buy expensive software that does not meet their needs sufficiently.

In computer science, there are many algorithms that finds a minimum spanning tree for a connected weighted undirected fuzzy graph. The minimum length (or cost) spanning tree problem is one of the nicest and simplest problems in network optimization, and it has a wide variety of applications. The problem is tofind a minimum cost (or length) spanning tree in G. Applications include the design of various types of distribution networks in which the nodes represent cities, centers etc.; and edges represent communication links (fiber glass phone lines, data transmission lines, cable TV lines, etc.), high voltage power transmissionlines, natural gas or crude oil pipelines, water pipelines, highways, etc. The objective is to design a network that connects all the nodes using the minimum length of cable or pipe or other resource in this paper we find the solution to the problem is to minimize the amount of new telephone line connection using matrixalgorithm with fuzzy graph.

The shortest path problem is considered as one of the essential problems in network optimization with a wide range of real-world applications. Modelling such real-world applications involves various indeterminate phenomena which can be estimated through human beliefs. The uncertainty theory proposed by Liu (Uncertain theory, 2nd edn., Springer, Berlin, 2007) is widely regarded as a legitimate tool to deal with such uncertainty. This paper presents an uncertain multi-objective shortest path problem (UMSPP) for a weighted connected directed graph (WCDG), where every edge weight is associated with two uncertain parameters: cost and time. These parameters are represented as uncertain variables. Here, we have formulated the expected value model and chance-constrained model of the proposed UMSPP, and the corresponding deterministic transformation of these models is also presented. Subsequently, the deterministic models are solved with a classical multi-objective solution method, namely the global criterion method. Furthermore, two multi-objective genetic algorithms (MOGAs): nondominated sorting genetic algorithm II (NSGA-II) and multi-objective cross-generational elitist selection, heterogeneous recombination and cataclysmic mutation (MOCHC), are employed to solve these models. A suitable example is provided to illustrate the proposed model. Finally, the performance of MOGAs is compared for five randomly generated instances of UMSPP.