寻求任意无向图的最大团集的一个算法及其程序

本文给出了一个有效的求任意简单图的最大团集(最大独立集,最小点覆盖)算法并给出了具体程序(用BASIC语言)。其时间复杂度为O(n~5)。     

一种改进的最大团问题DNA计算机算法(英文)

随着DNA计算的不断发展,如何克服穷举算法带来的指数爆炸问题已成为DNA计算领域的重要研究目标之一．将图灵机中的剪枝算法设计技术应用于最大团问题的DNA计算中,提出一种最大团问题的新DNA计算机算法．算法由顶点度数搜索器、团生成器、稀疏图与稠密图并行搜索器以及最大团搜索器组成．与已有文献同类算法的对比分析表明：文中算法在保持多项式操作时间的条件下,将求解n个顶点的最大团问题所需DNA分子链数从现有文献的O（2^n）减少至O（√3^n）,同时文中算法还具有高效的空间利用率及容错能力的优点．     

关于求平面点集凸包的一个O(n)时间算法的商榷

王志强等于1998年提出了一个计算平面点集凸包的新算法，并且声称该算法的最坏时间复杂度为O(n)，从而为张性时间排序提供了可能性，该文对王志强等提出的求平面点集凸包算法的时间分析提出了不同观点，进一步明确了平面集凸包算法和排序算法的时间下界为Ω(nlogn).     

高维空间球集覆盖问题的改进1+ε近似算法

高维空间球集的覆盖问题是指对高维空间中多个球构成的集合S,构造一个直径最小的球来覆盖S中所有已知球。本文提出了球集直径的概念,给出求解球集直径的1/3~(1/2)近似算法。基于此算法求解球集实例集合S的初始核心集,进而给出高维空间球集覆盖问题的1+ε近似算法,算法时间复杂度为O(nd/ε+d2/ε3/2(1/ε+d)lg1/ε)。算法保证核心集中球的个数为O(1/ε),与S中球的个数和空间维数无关。     

一种基于链暗示技术的Min-CVCB问题的精确算法

随着VLSI(超大规模集成电路)技术的发展,关于可重构阵列二分图的受约束最小点覆盖(Min-CVCB)问题受到了很多文献的关注.作为点覆盖问题的子问题,该问题已被证明是NP-完全问题.人们利用核心化和分支即使给出了时间复杂度为O((ku k1)|G| 1.26ku k1)的目前最好算法,然而仍不能满足实际工程的需要.通过进一步深入分析二分图的结构,对含有权值大于或等于3的块的连通子图分析其可能连接情况后充分利用"链暗示"技术和分枝搜索技术来建立起新的搜索递推关系;对于分枝后的块提出了一种动态规划算法,其可在多项式时间内完成处理.整个参数算法的运行时间为O((ku k1)|G| 1.1892ku k1),极大地改进了目前的最好结果.     

低度图的最大团求解算法

在图的最大团问题中,当图的顶点数不大于阈值m时,很容易求解其最大团问题,求解算法的时间复杂度为O(d)。给出一种求解低度图的最大团的确定性算法。该算法通过对图按顶点逐步分解实现分别计算,较好地解决低度图的最大团问题。算法时间复杂度为O(d•n3)。其中,n表示图的顶点数,图中顶点的最大度小于m或者图可以通过逐个删除度小于m的顶点而使所有顶点的度都小于m。     

分布式最小连通支配集启发式算法

针对AdHoc网络中用洪泛法进行广播易引起广播风暴的问题,提出一个新的分布式最小连通支配集启发式算法HMCDS,其中包括构建极大独立集、引入节点的有效度概念、选择有效度最大的节点作为支配点的贪心策略的方法,实验结果证明,HMCDS算法生成的连通支配集大小为7．60pt＋1．4,时间复杂度为O（△^2）,消息复杂度为O（n）,比同类算法优秀。     

高维空间球集覆盖问题的改进1+ε近似算法

高维空间球集的覆盖问题是指对高维空间中多个球构成的集合S,构造一个直径最小的球来覆盖S中所有已知球。本文提出了球集直径的概念,给出求解球集直径的1/〖KF(〗3〖KF)〗近似算法。基于此算法求解球集实例集合S的初始核心集,进而给出高维空间球集覆盖问题的1+ε近似算法,算法时间复杂度为O(nd/ε+d2/ε〖SX(〗3〖〗2〖SX)〗(1/ε+d)lg1/ε)。算法保证核心集中球的个数为 O(1/ε),与S中球的个数和空间维数无关。     

求解图的最大独立集的一种算法

如何寻找图的最大独立集这个问题是一个古老的难题。文章从图论的基本概念入手 ,得到了一种基于图的邻接矩阵的寻找图的极大独立集和最大独立集的算法 ,并得到其算法复杂度为 O(nn!/(m!(n - m) !) )     

测试集问题的集合覆盖贪心算法的深入近似

测试集问题是一个有着广泛应用的NP难问题.集合覆盖贪心算法是测试集问题的一个常用近似算法,其由集合覆盖问题得到的近似比21nn+1能否改进是一个公开的问题.集合覆盖贪心算法的推广被用来求解生物信息学中出现的冗余测试集问题.通过分析条目对被区分次数的分布情况,用去随机方法证明了集合覆盖贪心算法对测试集问题的近似比可以为1.51nn+0.5lnlnn+2,从而缩小了这种算法近似比分析的间隙.另外,给出了集合覆盖贪心算法对冗余度为n-1的加权冗余测试集问题的近似比的紧密下界(2-o(1))lnn-Θ 1).     

Exponential Speedup of Fixed-Parameter Algorithms for Classes of Graphs Excluding Single-Crossing Graphs as Minors

We present a fixed-parameter algorithm that constructively solves the $k$-dominating set problem on any class of graphs excluding a single-crossing graph (a graph that can be drawn in the plane with at most one crossing) as a minor in $O(4^{9.55\sqrt{k}}n^{O(1)})$ time. Examples of such graph classes are the $K_{3,3}$-minor-free graphs and the $K_{5}$-minor-free graphs. As a consequence, we extend our results to several other problems such as vertex cover, edge dominating set, independent set, clique-transversal set, kernels in digraphs, feedback vertex set, and a collection of vertex-removal problems. Our work generalizes and extends the recent results of exponential speedup in designing fixed-parameter algorithms on planar graphs due to Alber et al. to other (nonplanar) classes of graphs.     

一种可重构阵列的最小瑕点覆盖算法

关于可重构阵列的瑕点覆盖问题受到了很多文献的关注，特别地，关于可重构阵列的最小瑕点覆盖问题等价于二分图的受约束最小点覆盖问题，并被证明是NP-完全问题。针对本问题提出的算法运行时间为O(1．19^k kn)，这里k为可替换行与列的数目，改进了原有的最好结果，其运行时间为0(1．266k kn)，较好地组合并扩展了研究参数计算的最新技术与经典匹配理论，且具有较好的实用价值。这是关于可重构阵列的最小瑕点覆盖问题算法又一较大的改进，也是目前最小点覆盖问题相关参数算法的较有意义的改进。     

A learning approach to the bandwidth multicolouring problem

In this article, a generalisation of the vertex colouring problem known as bandwidth multicolouring problem (BMCP), in which a set of colours is assigned to each vertex such that the difference between the colours, assigned to each vertex and its neighbours, is by no means less than a predefined threshold, is considered. It is shown that the proposed method can be applied to solve the bandwidth colouring problem (BCP) as well. BMCP is known to be NP-hard in graph theory, and so a large number of approximation solutions, as well as exact algorithms, have been proposed to solve it. In this article, two learning automata-based approximation algorithms are proposed for estimating a near-optimal solution to the BMCP. We show, for the first proposed algorithm, that by choosing a proper learning rate, the algorithm finds the optimal solution with a probability close enough to unity. Moreover, we compute the worst-case time complexity of the first algorithm for finding a 1/(1–?) optimal solution to the given problem. The main advantage of this method is that a trade-off between the running time of algorithm and the colour set size (colouring optimality) can be made, by a proper choice of the learning rate also. Finally, it is shown that the running time of the proposed algorithm is independent of the graph size, and so it is a scalable algorithm for large graphs. The second proposed algorithm is compared with some well-known colouring algorithms and the results show the efficiency of the proposed algorithm in terms of the colour set size and running time of algorithm.     

Exact Sampling from Perfect Matchings of Dense Regular Bipartite Graphs

We present the first algorithm for generating random variates exactly uniformly from the set of perfect matchings of a bipartite graph with a polynomial expected running time over a nontrivial set of graphs. Previous Markov chain results obtain approximately uniform variates for arbitrary graphs in polynomial time, but their general running time is Θ(n¹⁰ (ln n)²). Other algorithms (such as Kasteleyn's O(n³) algorithm for planar graphs) concentrated on restricted versions of the problem. Our algorithm employs acceptance/rejection together with a new upper limit on the permanent of a form similar to Bregman's theorem. For graphs with 2n nodes, where the degree of every node is γn for a constant γ, the expected running time is O(n^{1.5 + .5/γ}). Under these conditions, Jerrum and Sinclair showed that a Markov chain of Broder can generate approximately uniform variates in Θ(n^{4.5 + .5/γ} ln n) time, making our algorithm significantly faster on this class of graphs. The problem of counting the number of perfect matchings in these types of graphs is # P complete. In addition, we give a 1 + σ approximation algorithm for finding the permanent of 0–1 matrices with identical row and column sums that runs in O(n^{1.5 + .5/γ} (1/σ²) log (1/δ))$, where the probability that the output is within 1 + \sigma$ of the permanent is at least 1 – δ.     

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.     

A 2-approximation NC algorithm for connected vertex cover and tree cover

The connected vertex cover problem is a variant of the vertex cover problem, in which a vertex cover is additional required to induce a connected subgraph in a given connected graph. The problem is known to be NP-hard and to be at least as hard to approximate as the vertex cover problem is. While several 2-approximation NC algorithms are known for vertex cover, whether unweighted or weighted, no parallel algorithm with guaranteed approximation is known for connected vertex cover. Moreover, converting the existing sequential 2-approximation algorithms for connected vertex cover to parallel ones results in RNC algorithms of rather high complexity at best.In this paper we present a 2-approximation NC (and RNC) algorithm for connected vertex cover (and tree cover). The NC algorithm runs in O(log²n) time using O(Δ²(m+n)/logn) processors on an EREW-PRAM, while the RNC algorithm runs in O(logn) expected time using O(m+n) processors on a CRCW-PRAM, when a given graph has n vertices and m edges with maximum vertex degree of Δ.     

Matching Algorithms Are Fast in Sparse Random Graphs

We present an improved average case analysis of the maximum cardinality matching problem. We show that in a bipartite or general random graph on n vertices, with high probability every non-maximum matching has an augmenting path of length O(log n). This implies that augmenting path algorithms like the Hopcroft-Karp algorithm for bipartite graphs and the Micali-Vazirani algorithm for general graphs, which have a worst case running time of O(m√n), run in time O(m log n) with high probability, where m is the number of edges in the graph. Motwani proved these results for random graphs when the average degree is at least ln (n) [Average Case Analysis of Algorithms for Matchings and Related Problems, Journal of the ACM, 41(6):1329-1356, 1994]. Our results hold if only the average degree is a large enough constant. At the same time we simplify the analysis of Motwani.     

随机图的点可区别V-全染色算法

图[G]的点可区别V-全染色就是相邻的边、顶点与其关联边必须染不同的颜色,同时要求所有顶点的色集合也不相同,所用的最少颜色数称为图[G]的点可区别V-全色数。根据点可区别V-全染色的约束规则,设计了一种启发式的点可区别V-全染色算法,该算法借助染色矩阵及色补集合逐步迭代交换,每次迭代交换后判断目标函数值,当目标函数值满足要求时染色成功。给出了算法的详细描述、算法分析和算法测试结果,对给定点数的图进行了点可区别V-全染色猜想的验证。实验结果表明,该算法有很好的执行效率并可以得到给定图的点可区别V-全色数,并且算法的时间复杂度不超过[O(n3)]。