Approximation algorithm for coloring of dotted interval graphs

Dotted interval graphs were introduced by Aumann et al. [Y. Aumann, M. Lewenstein, O. Melamud, R. Pinter, Z. Yakhini, Dotted interval graphs and high throughput genotyping, in: ACM-SIAM Symposium on Discrete Algorithms, SODA 2005, pp. 339-348] as a generalization of interval graphs. The problem of coloring these graphs found application in high-throughput genotyping. Jiang [M. Jiang, Approximating minimum coloring and maximum independent set in dotted interval graphs, Information Processing Letters 98 (2006) 29-33] improves the approximation ratio of Aumann et al. [Y. Aumann, M. Lewenstein, O. Melamud, R. Pinter, Z. Yakhini, Dotted interval graphs and high throughput genotyping, in: ACM-SIAM Symposium on Discrete Algorithms, SODA 2005, pp. 339-348]. In this work we improve the approximation ratio of Jiang [M. Jiang, Approximating minimum coloring and maximum independent set in dotted interval graphs, Information Processing Letters 98 (2006) 29-33] and Aumann et al. [Y. Aumann, M. Lewenstein, O. Melamud, R. Pinter, Z. Yakhini, Dotted interval graphs and high throughput genotyping, in: ACM-SIAM Symposium on Discrete Algorithms, SODA 2005, pp. 339-348]. In the exposition we develop a generalization of the problem of finding the maximum number of non-attacking queens on a triangle.     

1.5-Approximation algorithm for weighted maximum routing and wavelength assignment on rings

We give a 1.5-approximation algorithm for the weighted maximum routing and wavelength assignment problem on undirected ring networks. This improves the previous 1.58-approximation result.     

Weighted and unweighted maximum clique algorithms with upper bounds from fractional coloring

The linear programming relaxation of the minimum vertex coloring problem, called the fractional coloring problem, is NP-hard. We describe efficient approximation procedures for both the weighted and unweighted versions of the problem. These fractional coloring procedures are then used for generating upper bounds for the (weighted or unweighted) maximum clique problem in the framework of a branch-and-bound procedure. Extensive computational testing shows that replacing the standard upper bounding procedures based on various integer coloring heuristics with the somewhat more expensive fractional coloring procedure results in improvements of the bound by up to one-fourth in the unweighted and up to one-fifth in the weighted case, accompanied by a decrease in the size of the search tree by a factor of almost two.This research was supported by the National Science Foundation under Grant No. DDM-9201340 and the Office of Naval Research through Contract N00014-85-K-0198.     

Acyclic coloring of graphs of maximum degree five: Nine colors are enough

An acyclic coloring of a graph G is a coloring of its vertices such that: (i) no two neighbors in G are assigned the same color and (ii) no bicolored cycle can exist in G. The acyclic chromatic number of G is the least number of colors necessary to acyclically color G. In this paper, we show that any graph of maximum degree 5 has acyclic chromatic number at most 9, and we give a linear time algorithm that achieves this bound.     

Best Possible Approximation Algorithm for MAX SAT with Cardinality Constraint

We consider the MAX SAT problem with the additional constraint that at most P variables have a true value. We obtain a (1-e ^-1 ) -approximation algorithm for this problem. Feige [6] has proved that for MAX SAT with cardinality constraint with clauses without negations this is the best possible performance guarantee unless P=NP . Received August 20, 1998; revised June 23, 1999, and April 17, 2000.     

图的点可区别边染色算法研究

针对一般图设计了一种新型的点可区别边染色算法。该算法把概率思想和图染色相结合, 根据点可区别边染色的约束规则确立目标函数, 利用交换规则逐步寻优, 当目标函数的值满足要求时染色成功。给出详细算法步骤并进行了测试和分析, 实验结果表明该算法可以求出满足猜想的点可区别边色数。     

NIC-平面图中的轻边存在性及其定向染色

如果一个图[G]画在平面上有交叉[c],则该交叉可以与产生它的两条边所关联的4个顶点所构成的点集合[{v1,v2,v3,v4}]建立一个对应关系[θ:c→{v1,v2,v3,v4}]。如果对于[G]中任何两个不同的交叉（如果存在的话）[c1]与[c2]都有[|θ(c1)?θ(c2)|≤1],则称图[G]为NIC-平面图。证明了每个围长至少为5且最小度为4的NIC-平面图含有一条边,其2个顶点的度数都是4,从而每个围长至少为5的NIC-平面图的定向染色数至多为67。     

An improved approximation algorithm for the metric maximum clustering problem with given cluster sizes

The input to the metric maximum clustering problem with given cluster sizes consists of a complete graph G=(V,E) with edge weights satisfying the triangle inequality, and integers c₁,…,cp. The goal is to find a partition of V into disjoint clusters of sizes c₁,…,cp, maximizing the sum of weights of edges whose two ends belong to the same cluster. We describe an approximation algorithms for this problem with performance guarantee that approaches 0.5 when the cluster sizes are large.     

A logarithmic approximation algorithm for the minimum energy consumption broadcast subgraph problem

Motivated by the problem of supporting energy-efficient broadcasting in ad hoc wireless networks, we study the Minimum Energy Consumption Broadcast Subgraph (MECBS) problem. We present the first logarithmic approximation algorithm for the problem which uses an interesting reduction to Node-Weighted Connected Dominating Set.     

基于着色算法的并行碰撞检测算法

提出了一种基于着色算法的并行碰撞检测算法,利用AABB包围盒较好的紧密性和包围球计算简单的优点以及并行算法中的分治策略构建物体的混合包围体层次(S-AABB);然后采用破对称技术中的典型算法——着色算法,将每棵任务树编码,以产生各不相同的类别,并将不同的类别指派到不同的并行机,在并行机上采用多线程技术执行相同的类别的任务树的遍历,来检测是否有碰撞发生。实验结果表明,与现有的经典的I-COLLIDE等算法相比,该算法在效率、精确性方面具有明显优势,能够满足交互式复杂虚拟环境的实时性和精确性的要求。     

Flexible coloring

Motivated by reliability considerations in data deduplication for storage systems, we introduce the problem of flexible coloring. Given a hypergraph H and the number of allowable colors k, a flexible coloring of H is an assignment of one or more colors to each vertex such that, for each hyperedge, it is possible to choose a color from each vertex?s color list so that this hyperedge is strongly colored (i.e., each vertex has a different color). Different colors for the same vertex can be chosen for different incident hyperedges (hence the term flexible). The goal is to minimize color consumption, namely, the total number of colors assigned, counting multiplicities. Flexible coloring is NP-hard and trivially approximable, where s is the size of the largest hyperedge, and n is the number of vertices. Using a recent result by Bansal and Khot, we show that if k is constant, then it is UGC-hard to approximate to within a factor of s−ε, for arbitrarily small constant ε>0. Lastly, we present an algorithm with an approximation ratio, where k^′ is number of colors used by a strong coloring algorithm for H.     

An effective heuristic algorithm for sum coloring of graphs

Given an undirected graph G=(V,E), the minimum sum coloring problem (MSCP) is to find a legal vertex coloring of G, using colors represented by natural numbers (1,2,…) such that the total sum of the colors assigned to the vertices is minimized. In this paper, we present EXSCOL, a heuristic algorithm based on independent set extraction for this NP-hard problem. EXSCOL identifies iteratively collections of disjoint independent sets of equal size and assign to each independent set the smallest available color. For the purpose of computing large independent sets, EXSCOL employs a tabu search based heuristic. Experimental evaluations on a collection of 52 DIMACS and COLOR2 benchmark graphs show that the proposed approach achieves highly competitive results. For more than half of the graphs used in the literature, our approach improves the current best known upper bounds.     

一种基于主动生长的边缘连接算法*

针对因漏检而断裂的图像边缘图中的边缘连接问题,提出基于主动生长的边缘连接算法。先检测边缘图中端点的位置和方向,再在端点处进行主动边缘生长并通过退化操作实现边缘的正确连接;通过退化和约束生长实现对边缘图的去噪处理。实验表明,本算法在不增加边缘图噪声的情况下,很好地实现了对断裂边缘的有效连接,有利于后续的图像分割、表达和匹配等处理。     

基于追踪部署的着色包标记算法的研究 *

基于追踪部署的相关理论和着色包标记算法,针对当前危害很大的分布式拒绝服务攻击,提出一种基于追踪部署的IP回溯算法。该算法是以贪心算法为基础,利用K-剪枝算法在网络拓扑图中找出一些关键的路由器,利用这些路由器也就是只让tracers对过往的数据包按照着色包标记算法进行标记,这样不但减少了重构路径所需的数据包数,降低了路径误报率,提高了追踪到攻击者的速度,而且大大减轻了路由器标记的负担,从而能够迅速准确地找到攻击源。     

Cliques with maximum/minimum edge neighborhood and neighborhood density

Given a graph G=(V,E) and a clique C of G, the edge neighborhood of C can be defined as the total number of edges running between C and V?C, being denoted by N(C). The density of the edge neighborhood N(C) can be set as the ratio (|N(C)|/(|C|·|V?C|)).This paper addresses maximum/minimum edge neighborhood and neighborhood density cliques in G. Two versions will be undertaken, by fixing, or not, the size of the cliques.From an optimization point of view these problems do not bring much novelty, as they can be seen as particular or special versions of weighted clique problems. However, from a practical point of view, they concentrate on certain kinds of properties of cliques, rather than their size, revealing clique's engagement in the graph. In fact, a maximum edge neighborhood clique should be strongly embraced in the graph, while a minimum edge neighborhood clique should reveal an almost isolated strong component. In particular, special versions of the new problems allow to distinguish among cliques of the same size, namely among possible tied maximum cliques in the graph.We propose node based formulations for these edge based clique related problems. Using these models, we present computational results and suggest applications where the new problems can bring additional insights.     

基于追踪部署的着色包标记算法的研究 *

基于追踪部署的相关理论和着色包标记算法 ,针对当前危害很大的分布式拒绝服务攻击 ,提出一种基于追踪部署的 IP回溯算法。该算法是以贪心算法为基础 ,利用 K-剪枝算法在网络拓扑图中找出一些关键的路由器 ,利用这些路由器也就是只让 tracers对过往的数据包按照着色包标记算法进行标记 ,这样不但减少了重构路径所需的数据包数 ,降低了路径误报率 ,提高了追踪到攻击者的速度 ,而且大大减轻了路由器标记的负担 ,从而能够迅速准确地找到攻击源。     

Fully dynamic algorithms for permutation graph coloring

We introduce a dynamic model for maintaining permutation graph coloring. Our motivation comes from the strait type river routing problem in VLSI. This paper presents fully dynamic algorithms for the permutation graph coloring problem. These algorithms are designed to handle Insert and Delete operations and answer some queries. The aim is to provide for running times that are asymptotically more efficient than recomputation (off-line algorithms that run in 0(n logw) time, are known [5,6,10,3]). First, the algorithm A^ that runs in 0(n) uniform running time per Insert/Delete operation is presented. Second, a more sophisticated data structure leads to the algorithm A2 that runs in (9(m logw) uniform running time per Insert I Delete, where m denotes the number of chains in the decomposition. It follows from [7,4] that the running time of A2 when the points from the dynamically changing set are drawn independently from a uniform distribution on the unit square is G(yfn logn) per Insert/Delete in probability. Third, we sketch a composite algorithm A3 that switches between A± and A2 guarantees an amortized running time of (min{n,m logw)) per Insert/Delete. Finally, we outline a number of applications