A hybrid algorithmic model for the minimum weight dominating set problem

Iterated greedy algorithms belong to the class of stochastic local search methods. They are based on the simple and effective principle of generating a sequence of solutions by iterating over a constructive greedy heuristic using destruction and construction phases. This paper, first, presents an efficient randomized iterated greedy approach for the minimum weight dominating set problem, where—given a vertex-weighted graph—the goal is to identify a subset of the graphs’ vertices with minimum total weight such that each vertex of the graph is either in the subset or has a neighbor in the subset. Our proposed approach works on a population of solutions rather than on a single one. Moreover, it is based on a fast randomized construction procedure making use of two different greedy heuristics. Secondly, we present a hybrid algorithmic model in which the proposed iterated greedy algorithm is combined with the mathematical programming solver CPLEX. In particular, we improve the best solution provided by the iterated greedy algorithm with the solution polishing feature of CPLEX. The simulation results obtained on a widely used set of benchmark instances shows that our proposed algorithms outperform current state-of-the-art approaches.     

The complexity for partitioning graphs by monochromatic trees,cycles and paths

Let G be an edge-coloured graph. We show in this paper that it is NP-hard to find the minimum number of vertex disjoint monochromatic trees which cover the vertices of the graph G. We also show that there is no constant factor approximation algorithm for the problem unless P?=?NP. The same results hold for the problem of finding the minimum number of vertex disjoint monochromatic cycles (paths, respectively) which cover the vertices of the graph.     

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.     

最小赋权支配集的迭代禁忌搜索算法

最小赋权支配集是一个NP困难的组合优化问题,有着广泛的应用背景。提出了一个高效的求解最小赋权支配集的迭代禁忌搜索算法。该算法采用随机贪心构造算法构造初始解,并利用快速的局部禁忌搜索算法寻找局部最优解,通过随机扰动和修复策略来搜索新的区域,以期跳出当前的局部最优解。用顶点数为800到1 000的大规模标准测试例子测试提出的算法。数值实验结果和与现存的启发式算法比较结果表明了算法是有效的。     

基于邻接矩阵的网络流量检测点设置算法

在研究网络流量的有效测量问题时,考虑网络节点的流守恒,把网络流量监测点问题抽象为无向图的最小弱顶点覆盖问题,这是一个NP难的问题.基于图论中邻接矩阵的概念,提出一个近似算法,通过重复删除邻接矩阵中所有行元素之和不超过1的节点对应的行和列,得到最小弱顶点覆盖集.在此基础上通过预先递归去除无向图中1度节点,满足任意节点度数都大于或等于2的最小弱顶点覆盖问题求解条件,并将递归节点作为该近似算法的入口点.仿真实验表明,与现有算法相比,新算法具有更好的性能,能够发现更小的弱顶点覆盖集.     

A NEW APPROACH TO THE VERTEX COLORING PROBLEM

The vertex coloring problem is a well-known classical optimization problem in graph theory in which a color is assigned to each vertex of the graph in such a way that no two adjacent vertices have the same color. The minimum vertex coloring problem is known to be an NP-hard problem in an arbitrary graph, and a host of approximation solutions are available. In this article, a learning automata–based approximation algorithm is proposed to solve the minimum vertex coloring problem. The proposed algorithm iteratively finds the different possible colorings of the graph and compares it at each stage with the best coloring found so far. If the number of distinct colors in the chosen coloring is less than that of the best coloring, the chosen coloring is rewarded; otherwise, it is penalized. Convergence of the proposed algorithm to the optimal solution is proven. The proposed vertex coloring algorithm is compared with the well-known coloring techniques and the results show the superiority of the proposed algorithm over the others both in terms of the color set size and running time of algorithm.     

一种基于元胞自动机的无向图剖分优化算法

运用元胞自动机理论,针对无向图剖分优化问题进行了分析和建模,提出了一种元胞自动机模型以及基于该模型的无向图剖分优化算法。在该元胞自动机模型中,元胞对应于无向图中的结点,元胞的邻居对应于邻接结点,元胞空间对应于无向图中的结点集,元胞的状态对应于所在的结点子集。实验及分析表明该算法不仅能找到无向图的近似最优剖分,而且有效地降低了空间复杂度和时间复杂度。     

一种混合化学反应优化算法求解最小顶点覆盖问题

最小顶点覆盖问题是组合最优化问题,在实际应用中有较广泛的应用,是一个NP难问题。论文针对最小顶点覆盖问题给出了一种混合化学反应优化求解算法。首先根据无向图的邻接矩阵表示法,设计了参与化学化反应的分子编码和目标函数;同时把贪心算法思想创造性地融入到化学反应优化算法的四个重要反应算子中,以加快局部较优解的搜索过程;最后通过模拟化学反应中分子势能趋于稳定的过程,在问题的解空间中搜索其最优解。模拟实验结果表明,该算法对于求解无向图的最小顶点覆盖问题是有效的,并且在求解效率等方面有一定的改善。     

On the k-path cover problem for cacti

In this paper we investigate the k-path cover problem for graphs, which is to find the minimum number of vertex disjoint k-paths that cover all the vertices of a graph. The k-path cover problem for general graphs is NP-complete. Though notable applications of this problem to database design, network, VLSI design, ring protocols, and code optimization, efficient algorithms are known for only few special classes of graphs. In order to solve this problem for cacti, i.e., graphs where no edge lies on more than one cycle, we introduce the so-called Steiner version of the k-path cover problem, and develop an efficient algorithm for the Steiner k-path cover problem for cacti, which finds an optimal k-path cover for a given cactus in polynomial time.     

模糊环境下的最小权顶点覆盖问题*

最小权顶点覆盖问题在实际决策中应用广泛,但顶点上的权值在实际应用中通常代表费用、成本等,在很多情况下是不确定的。关注了最小权顶点覆盖问题中的模糊不确定性,对模糊环境下的最小权顶点覆盖问题进行了研究。引入了可信性理论以描述模糊不确定性,并根据不同的决策准则建立了求解模糊环境下最小权顶点覆盖问题的三个决策模型,结合模糊模拟和遗传算法设计了一种求解所建立模型的混合智能算法,并给出了数值实验。数值实验的结果验证了所提出的决策模型与算法的有效性。     

基于原始对偶方法求解网络流量监测集算法

考虑网络节点的流守恒特性,网络流量的有效监测问题可抽象为求给定图G(V,E)的最小弱顶点覆盖集的问题和基于流划分的最小弱顶点覆盖集的问题,这是NP难的问题.首先分析了弱顶点覆盖集的约束关系,并给出了问题的整数规划形式.然后利用原始对偶方法构造了求解最小弱顶点覆盖集的近似算法,并分析了算法的比界为2.进一步分析了求解基于最大流划分的最小弱顶点覆盖集的近似算法.     

An integer programming-based local search for the covering salesman problem

We consider a generalized version of the well known Traveling Salesman Problem called Covering Salesman problem. In this problem, we are given a set of vertices while each vertex i can cover a subset of vertices within its predetermined covering distance r_i. The goal is to construct a minimum length Hamiltonian cycle over a subset of vertices in which those vertices not visited on the tour has to be within the covering distance of at least one vertex visited on the tour. The paper proposes an Integer Linear Programming based heuristic method which takes advantage of Integer Linear Programming techniques and heuristic search to improve the quality of the solutions. Extensive computational tests on the standard benchmark instances and on a new set of large sized datasets show the effectiveness of the proposed approach.     

An 11/6-approximation algorithm for the network steiner problem

An instance of the Network Steiner Problem consists of an undirected graph with edge lengths and a subset of vertices; the goal is to find a minimum cost Steiner tree of the given subset (i.e., minimum cost subset of edges which spans it). An 11/6-approximation algorithm for this problem is given. The approximate Steiner tree can be computed in the time0(¦V¦ ¦E¦ + ¦S¦⁴), whereV is the vertex set,E is the edge set of the graph, andS is the given subset of vertices.     

Analysis of the $(1+1)$-EA for Finding Approximate Solutions to Vertex Cover Problems

Vertex cover is one of the best known NP-hard combinatorial optimization problems. Experimental work has claimed that evolutionary algorithms (EAs) perform fairly well for the problem and can compete with problem-specific ones. A theoretical analysis that explains these empirical results is presented concerning the random local search algorithm and the (1+1)-EA. Since it is not expected that an algorithm can solve the vertex cover problem in polynomial time, a worst case approximation analysis is carried out for the two considered algorithms and comparisons with the best known problem-specific ones are presented. By studying instance classes of the problem, general results are derived. Although arbitrarily bad approximation ratios of the (1+1)-EA can be proved for a bipartite instance class, the same algorithm can quickly find the minimum cover of the graph when a restart strategy is used. Instance classes where multiple runs cannot considerably improve the performance of the (1+1)-EA are considered and the characteristics of the graphs that make the optimization task hard for the algorithm are investigated and highlighted. An instance class is designed to prove that the (1+1)-EA cannot guarantee better solutions than the state-of-the-art algorithm for vertex cover if worst cases are considered. In particular, a lower bound for the worst case approximation ratio, slightly less than two, is proved. Nevertheless, there are subclasses of the vertex cover problem for which the (1+1)-EA is efficient. It is proved that if the vertex degree is at most two, then the algorithm can solve the problem in polynomial time.     

Polynomial-time algorithms for weighted efficient domination problems in AT-free graphs and dually chordal graphs

An efficient dominating set (or perfect code) in a graph is a set of vertices the closed neighborhoods of which partition the vertex set of the graph. The minimum weight efficient domination problem is the problem of finding an efficient dominating set of minimum weight in a given vertex-weighted graph; the maximum weight efficient domination problem is defined similarly. We develop a framework for solving the weighted efficient domination problems based on a reduction to the maximum weight independent set problem in the square of the input graph. Using this approach, we improve on several previous results from the literature by deriving polynomial-time algorithms for the weighted efficient domination problems in the classes of dually chordal and AT-free graphs. In particular, this answers a question by Lu and Tang regarding the complexity of the minimum weight efficient domination problem in strongly chordal graphs.     

一种增量式约简方法求解最小顶点覆盖问题

最小顶点覆盖问题是一个应用很广泛的NP难题,针对该问题给出一种增量式属性约简方法。首先将最小顶点覆盖问题转化为一个决策表的最小属性约简问题;利用增量式属性约简思想,随着图中边数的增多,提出一种更新最小顶点覆盖的增量式属性约简算法;该算法时间复杂度低于计算整个图的最小顶点覆盖的时间复杂度,同时针对大规模图问题,可随着边的增加动态更新最小顶点覆盖,因此降低了属性约简的方法求解最小顶点覆盖问题的运行时间;实验结果表明该算法的可行性和有效性。     

Degree constrained minimum spanning tree problem: a learning automata approach

Degree-constrained minimum spanning tree problem is an NP-hard bicriteria combinatorial optimization problem seeking for the minimum weight spanning tree subject to an additional degree constraint on graph vertices. Due to the NP-hardness of the problem, heuristics are more promising approaches to find a near optimal solution in a reasonable time. This paper proposes a decentralized learning automata-based heuristic called LACT for approximating the DCMST problem. LACT is an iterative algorithm, and at each iteration a degree-constrained spanning tree is randomly constructed. Each vertex selects one of its incident edges and rewards it if its weight is not greater than the minimum weight seen so far and penalizes it otherwise. Therefore, the vertices learn how to locally connect them to the degree-constrained spanning tree through the minimum weight edge subject to the degree constraint. Based on the martingale theorem, the convergence of the proposed algorithm to the optimal solution is proved. Several simulation experiments are performed to examine the performance of the proposed algorithm on well-known Euclidean and non-Euclidean hard-to-solve problem instances. The obtained results are compared with those of best-known algorithms in terms of the solution quality and running time. From the results, it is observed that the proposed algorithm significantly outperforms the existing method.     

网络流量的有效测量方法分析

把网络流量的有效测量问题抽象为求给定图G=(V,E)的最小弱顶点覆盖集的问题.给出了一个求最小弱顶点覆盖集的近似算法,并证明了该算法具有比界2(lnd+1),其中d是图G中顶点的最大度.指出了该算法的时间复杂性为O(|V|²).     

Experimental Analysis of Heuristic Algorithms for the Dominating Set Problem

We say a vertex v in a graph G covers a vertex w if v=w or if v and w are adjacent. A subset of vertices of G is a dominating set if it collectively covers all vertices in the graph. The dominating set problem, which is NP-hard, consists of finding a smallest possible dominating set for a graph. The straightforward greedy strategy for finding a small dominating set in a graph consists of successively choosing vertices which cover the largest possible number of previously uncovered vertices. Several variations on this greedy heuristic are described and the results of extensive testing of these variations is presented. A more sophisticated procedure for choosing vertices, which takes into account the number of ways in which an uncovered vertex may be covered, appears to be the most successful of the algorithms which are analyzed. For our experimental testing, we used both random graphs and graphs constructed by test case generators which produce graphs with a given density and a specified size for the smallest dominating set. We found that these generators were able to produce challenging graphs for the algorithms, thus helping to discriminate among them, and allowing a greater variety of graphs to be used in the experiments. Received October 27, 1998; revised March 25, 2001.     

Minimum cost source location problem with vertex-connectivity requirements in digraphs

Given a digraph (or an undirected graph) G=(V,E) with a set V of vertices v with nonnegative real costs w(v), and a set E of edges and a positive integer k, we deal with the problem of finding a minimum cost subset SV such that, for each vertex vV−S, there are k vertex-disjoint paths from S to v. In this paper, we show that the problem can be solved by a greedy algorithm in time in a digraph (or in time in an undirected graph), where n=|V| and m=|E|. Based on this, given a digraph and two integers k and ℓ, we also give a polynomial time algorithm for finding a minimum cost subset SV such that for each vertex vV−S, there are k vertex-disjoint paths from S to v as well as ℓ vertex-disjoint paths from v to S.